\section{Preliminars.}
The paper is organized as follows. In the first part, we revise a few concepts of convex geometry, the Gordan lemma and separation lemma, as important preliminary results for the subsequent developments as Hilbert basis. In the second section we revise the standard theory of toric algebraic geometry, \cite{BSturmfels1996}, \cite{DCoxLittleSchenck2010}, \cite{GEwald1993}; and make use of the definition of toric variety as an algebraic affine scheme, a definition that will permit the formalizations we show for singular machines and S-systems. In the third section, we enunciate a proof of the Hilbert basis lemma and we compute toric ideals. The value of this result and the computing by means of the Singular program, ref., \cite{DGPS}, enable us to compute  toric ideals as the basis for applications in statistical learning. Furthermore, we define toric morphisms and gluing maps which are of the great importance for the proof of theorem 1. These applications give evidence of the relevance of theorem 1 and its potential benefit to facilitating solutions of problems in engineering.\\
We also give a formal definition of singularity, Ewald \cite{GEwald1993}, and enunciate two theorems for toric resolution; one of them is the theorem of Atiyah-Hironaka; S. Watanabe, \cite{SWatanabe12001}, which is applied for the resolution of singularities due to S. Watanabe, \cite{SWatanabe12001}, \cite{SWatanabe32001}, \cite{SWatanabe62001}, this fact is our motivation to study toric varieties in singular machines and embedding of its parameter space associated, into projective spaces as the theorem 3 proves.\\
In the fourth section, we study and summarize the main concepts of statistical singular learning (identifiable and non identifiable machines, Kullback distance, Fisher matrix information, learning curve and singular machines) with the purpose of making a formal study of singular machines by means of toric resolutions and affine toric varieties where we enunciate and prove part of the Theorem 6 applying the results of the first part. We also see the effect of the singularities in statistical learning and its importance for the performance and training in singular machines, \cite{SWatanabe32001}, which is resolved and studied by means of Theorem 6. We conclude this section with applications for three different statistical machines (perceptron of two layers, mix of binomial distributions, and three layer perceptron) and compute the learning curves by means of Hilbert basis reproducing the results of S. Watanabe, \cite{SWatanabe52001},\cite{SWatanabeYamazaki2002}, \cite{KYamazakiWatanabe2004}.

\section{Background of Convex Combinatorial Geometry.}
All this compilation of definitions and concepts can be consulted in; G. Ewald, \cite{GEwald1993}.
A set   $ S \, \subset \, \mathbb{R}^{n}\setminus \emptyset $ is a  convex set if   each  $ \alpha \,\in  S $ is a convex  combination of elements of $S$; that is,   $\displaystyle \alpha= \sum_{i=1}^{r} \lambda_{i} \alpha_{i}$, where $ \lambda_{i} \geq \, 0  $  and $\alpha_i \in S$  for all $i =1,\dots, r$, with       $ \displaystyle \sum_{i=1}^{r} \lambda_{i}=1 $. \\
Given  $M \subset \mathbb{R}^{n} $, by \textbf{conv}$ M $ we mean the \textbf{hull convex} of $M$, which is the set of all convex combinations of elements of $M$. Moreover, if $ M $ is a finite set then \textbf{conv}$ M $ is called  a \textbf{convex polytope} or    \textbf{polytope}. \\
 A \textbf{lattice} $ N $ is a free abelian group of finite rank, and if its rank is $ n \in\mathbb N $, then $N$ is isomorphic to $ \mathbb{Z}^{n} $.\\
 Let  $ M $ and $ N $ be two lattice  both of rank $n$, consider $\:  \langle \cdot, \cdot \rangle  :  M \times N   \,  \longrightarrow \, \mathbb{Z} $,  the usual homomorphism of lattice from the inner product in $\mathbb R^n$ and identify to $ N $ with $ Hom_{\mathbb{Z}}(M,\mathbb{Z}) $, then we say
that $ N $ is the dual lattice of the lattice $ M $, and reciprocally. In any case one denotes $ N=M^{\vee} $, see for more details of this formalism \cite{DCoxLittleSchenck2010}.\\
 Given  $ M $ and $ N $  as dual lattice, denote  $ M_{\mathbb{R}} = M\otimes_{Z}\mathbb{R}$ and $ N_{\mathbb{R}} = N\otimes_{Z}\mathbb{R} $, and  set  $ \sigma =Con( S  ) \subseteq \, M_{\mathbb{R}} $,  for some set $ S \, \subseteq \, M $, then $ \sigma $ is called a \textbf{rational polyhedric cone} or  \textbf{lattice cone}, \cite{DCoxLittleSchenck2010}.\\
 Also a \textbf{lattice cone} is a cone $ \sigma= Con(\alpha_{1},...,\alpha_{r}) \subset \, \mathbb{Z}^{n} $, generated by, $\alpha_{1},...,\alpha_{r}  \,  \in  \mathbb{Z}^{n}$ vectors. If the  coordinates of $\alpha_i$ are relative primes to pairs for each $i=1,\dots,r$, then $\alpha_{1},...,\alpha_{r} $ are called \textbf{primitive vectors}, and the cone $ \sigma $ is called a \textbf{regular cone}. It is well known that if $ \alpha_{1},...,\alpha_{r}  $, are primitives, then there exists $ \alpha_{r+1},...,\alpha_{n} \, \in \mathbb{Z}^{n}  $ such that:
\begin{center}
\textbf{Det}$ (\alpha_{1},...,\alpha_{n})= \pm 1 $.
\end{center}
Also, if the $ \alpha_{1},...,\alpha_{n} \, \in \mathbb{Z}^{n} $ are linearly independent, then the cone $ \sigma $ is a \textbf{simplex cone or simplicial cone.}\\
 A \textbf{face}, $ \tau $ of a cone $ \sigma $ is $ H_{p}\cap \sigma $ where $ H_{p} \subset \, \mathbb{R}^{n} $  is a  tangent hyperplane   to $\sigma$ at  $ p \in \sigma $, it is usually denoted by $ \tau \, \preceq \, \sigma $ it is well known $ \preceq $ is a relation of order.\\
 The   relative interior of $ \sigma $ is   $Relint(\sigma)= \sigma^{\vee}  \setminus    \sigma^{\perp}   $, where
\begin{center}
$\sigma^{\vee} = \lbrace m \in \sigma: \langle m,u \rangle > 0, \, \forall u \, \in \sigma \rbrace $,\\
$ \sigma^{\perp}= \lbrace m \in \mathbb{R}^{n}: \langle m,u \rangle =0, \, \forall u \, \in \sigma  \rbrace$.
\end{center}
  Let  $ P \, \subseteq \, M_{\mathbb{R}} $  be a lattice polytope. A set of cones $ \sum_{F} = \lbrace \sigma_{F} \, \vert \, F \, \preceq \,  P \rbrace$,  is called a \textbf{Fan} if and only if:
	\begin{itemize}
	\item  If  $\tau \preceq \sigma_{F}  $, then $ \tau \preceq \sum_{F}  $  for each  $\sigma_{F} \, \in \, \sum_{F} $,   .\\
\item  If $\tau = \sigma_{1}\cap \sigma_{2} $, then  $ \tau \, \preceq \, \sigma_{1} $ and $ \tau \, \preceq \, \sigma_{2} $ for each $
\sigma_{1}, \sigma_{2}  \, \in \,  \sum_{F}$.
 \end{itemize}
Recalling the following facts:
\begin{enumerate}
\item Separation Lemma: If $ \sigma_{1} $ and $ \sigma_{2} $ are lattice cones in $ M $, whose intersection $ \tau =\sigma_{1} \cap \sigma_{2} $ is a face of both, then there is exists a hyperplane $ H_{m} $ such that:
\begin{center}
$ \tau =\sigma_{1} \cap H_{m}=\sigma_{2} \cap H_{m}$,
\end{center}
for any $ m \, \in \, Relint(\sigma)= \sigma_{1}^{\vee} \cap (-\sigma_{2})^{\vee}$.\\
\item Lemma. Set $\tau \, \preceq \, \sigma$ and $ m \,\in Relint(\tau^{\perp} \cap \sigma^{\vee} ) \setminus   \lbrace 0 \rbrace $. Then
\begin{center}
$ \tau^{\perp}= \sigma^{\vee} \oplus\{ \lambda (-m)\ \mid \  \lambda \, \in \, \mathbb{R} \}$.
\end{center}
\item Lemma. Let $ \sigma \, \subset \, \mathbb{R}^{n} $ be a lattice cone, then $ \sigma \cap \mathbb{Z}^{n} $ is a \textbf{monoid}.
\item Gordan lemma. Let $ \sigma \, \subset \, \mathbb{R}^{n} $ be a lattice cone,   then the monoid $ \sigma \cap \mathbb{Z}^{n} $ is  finitely generated.
\item Theorem. Let $ \sigma \subset \, \mathbb{R}^{n} $ be a  $ n $ dimensional  lattice cone   with \'{a}pex $ 0 $, i.e.; $0 \preceq \sigma $, and let $ b_{1},...,b_{r} $ be  the inner normal facets of $ \sigma $. Then
\begin{center}
$  \sigma^{\vee}= Con(b_{1},...,b_{r})$.
\end{center}
\end{enumerate}

\subsection{About toric algebraic geometry.}

 The affine variety $(\mathbb{C}^{*})^{n} =(\mathbb{C}/ \lbrace 0 \rbrace)^{n} $ is a group equipped  with the  complex product of coordinates to pairs and it is  called the \textbf{complex algebraic  n-torus}. A \textbf{torus T} is an affine variety isomorphic to $(\mathbb{C}^{*})^{n}$.\\
 A \textbf{character} of a torus \textbf{T} is a homomorphism of groups, $ \chi  : T\longrightarrow \mathbb{C}^{*} $. For example,  set  $ m=(a_{1},...,a_{n}) \, \in \mathbb{Z}^{n} $, then $ \chi^m  :(\mathbb{C}^{*})^{n} \longrightarrow \mathbb{C}^{*} $   given by:
\begin{center}
$  \chi^{m}(t_{1},...,t_{n})=t_{1}^{a_{1}}*...*t_{n}^{a_{n}} $,
\end{center}
is \textbf{character} of $(\mathbb{C}^{*})^{n} $. Even more, it is well known that any character of $(\mathbb{C}^{*})^{n}$ is given  as above.
Note that given  a lattice $ M $  and $m\in M$, then it is possible to define a character of $T$ by  $ \chi^{m}:T \longrightarrow \mathbb{C}^{*} $.\\

By an \textbf{uni-parametric subgroup} of a torus \textbf{T}  we mean  a homomorphism of groups $ \lambda : \mathbb{C}  ^{*} \longrightarrow  T$. Given  $ u =(b_{1},...,b_{n}) \, \in \mathbb{Z}^{n}$  define $ \lambda^{u}: \mathbb{C} ^{*} \longrightarrow  (\mathbb{C}^{n})^{*} $ by:
\begin{center}
$ \lambda^{u}(t_{1},...,t_{n})=(t_{1}^{b_{1}}, \dots, t_{n}^{b_{n}})$.
\end{center}
Then $\lambda^{u}$
 is  a uni-parametric subgroup of $(\mathbb{C}^{n})^{*}$ and any uni-parametric subgroup of $(\mathbb{C}^{n})^{*}$ is given in the same form.

One sees that  given a Torus \textbf{T},  there holds that   all uni-parametric subgroups of   $T$ form a free abelian group $ N $ with the same dimension of \textbf{T}. The same fact is obtained for all  characters of  \textbf{T}.

 The ring:
\begin{center}
$  \mathbb{C}[t,t^{-1}]=\mathbb{C}[t_{1},...,t_{n},t_{1}^{-1},...,t_{n}^{-1}] $
\end{center}
is called the \textbf{ ring of Laurent polynomials} and the monomials,
\begin{center}
$  \lambda * t^{a}= \lambda *t^{a_{1}}*...*t^{a_{n}}$ with $ a=(a_{1},...,a_{n}) \, \in \mathbb{Z}^{n}, \, \lambda \, \mathbb{C^{*}}$.
\end{center}
are called  \textbf{Laurent monomials}.\\

The \textbf{support} of  a Laurent polynomial  $ f=\sum_{i=1}^{r}\lambda_{i}t^{a_{i}} $,  is
\begin{center}
\textbf{supp}(f)$ =\lbrace a_i \, \in \, \mathbb{Z}^{n}: \, \lambda_{i} \neq 0 \rbrace$.
\end{center}

It is known that given $\mathbb{C}[t,t^{-1}] $ as above, and  let $ \sigma $  be lattice cone. Then
\begin{center}
$ R_{\sigma}=\lbrace f \, \in \, \mathbb{C}[t,t^{-1}] :$ \textbf{supp} $ (f) \subset \, \sigma \rbrace$
\end{center}
  is a  generated finitely  monomial  $ \mathbb{C} -$algebra.\\

\begin{definition}\label{Definition.} An \textbf{affine toric variety} is an irreducible affine variety $ X $ containing  a torus $ T_{N}\simeq (\mathbb{C^{*}})^{n}  $ as Zariski open  subset, such that the action of $ T_{N} $ on itself,  is extended to an algebraic action of $ T_{N} $ on $ X $; that is, there exists   a morphism from $ T_{N} \times X$ to $ X $, \cite{DCoxLittleSchenck2010} .\\

Let $\sigma$  be a lattice cone, the
   \textbf{affine algebraic scheme:}
\begin{center}
$  X_{\sigma}=$ \textbf{Spec} $ (R_{\sigma}) $.
\end{center}
  is called \textbf{abstract toric affine variety} or \textbf{embedding of torus.}\\
\end{definition}
For example, set $ 0\leq \, r \, \leq n $, and let   $ \sigma \subset \, \mathbb{R}^{n}$ be a lattice cone generated as follows $ \sigma= Con(e_{1},...,e_{r}) $ where  $ e_{i} $ are  canonical vectors in $ \mathbb{R}^{n} $ for  $ i=1,...,r $. Then computing  its dual cone, one has  $ \sigma^{\vee}= Con(e_{1},...,e_{r}, \pm e_{1},...,\pm e_{n}) $,  and  the   affine toric variety is
\begin{center}
$  X_{\sigma}=$ \textbf{Spec} $ \mathbb{C}[t_{1},...,t_{r},t_{r+1}^{\pm 1},...,t_{r+n}^{\pm 1}] \simeq \mathbb{C}^{r} \times (\mathbb{C}^{*})^{n-r}$.
\end{center}
this example is seen in, \cite{DCoxLittleSchenck2010}.
\section{Hilbert Basis.}
The theory of \textbf{Hilbert basis} is an important  algebraic geometry tool. The major contribution of this work
is  the employment of \textbf{Hilbert basis} associated to a monoid $ N \subset \mathbb{Z}^{n} $ to give explicitly a toric resolution  in  a  set  of  new   coordinates to solve a problem  in singular statistical learning in an original way; also see; \cite{BSturmfels1996}.
\\

Let  $ N \simeq \mathbb{Z}^{n}$ be the lattice  and set $ M= N^{\vee} $ its dual lattice. Let  $ \sigma $ be  a lattice cone defined in $ N $ and let $ \sigma^{\vee} $ be its dual cone  in $ M $. Denote   $ S_{\sigma} = \sigma^{\vee} \cap M$ and note that his monoid is finitely generated  (see Gordan lemma).

  \begin{lemma}\label{Lemma 2.1.1.}  (Basis Hilbert). Set $ \sigma \subseteq \, N $, then $\sigma$   is a   n-dimensional  cone  if and only if it is a strongly convex cone; i.e.,  $ \sigma\cap (-\sigma^{\vee}) = \lbrace 0 \rbrace$. In this case the monoid $ S_{\sigma} $ has a finite minimal set of generators \textbf{H} $ \subseteq \, M \simeq \mathbb{Z}^{d}$ and these are minimal, for details of this proof, see  \cite{DCoxLittleOshea2004},  \cite{DCoxLittleSchenck2010}.\end{lemma}

\begin{definition}\label{Definition 2.1.1} Set $ \omega = (\omega_{1},...,\omega_{n}) \, \in \mathbb{R}^{n} $, and for a polynomial $ f = \sum_{i=1}^n \lambda_{i}t^{a_{i}} $   define its  \textbf{initial form}  $ in_{\omega} (f)$, as the sum over all lambda terms, such that the inner product $ \langle \, \omega, a_{i} \, \rangle $ is maximal. \\
For an ideal $ I $, we mean  the \textbf{initial ideal} as the ideal generated from the initial forms
\begin{center}
$ in_{\omega} (I) =  \langle \, in_{\omega}(f) : f \, \in I \, \rangle$.
\end{center}
\end{definition}
\begin{definition}\label{Definition 2.1.2.} Each  polynomial $ f = \sum_{i =1}^{n}\lambda_{i}t^{a_{i}} $,
in the ring $ \mathbb{C}[t, t^{-1}] $,  is associated to a  convex polytope, or convex hull, in $ \mathbb{R}^{n} $ as follows:
\begin{center}
$  New(f)= $ \textbf{Conv} $ \lbrace a_{i} : i=1,...,m \rbrace \, \subset \mathbb{R}^{n} $.
\end{center}
$ New(f)$ is called the \textbf{Newton polytope} associated to $ f $, in the literature of singular learning machine it is known as the \textbf{exponent space}. But, generally, it is called  the exponent space generated by a Newton polytope; see ref., \cite{BSturmfels1996}.\\
\end{definition}
\begin{lemma}\label{Lemma 2.1.2.} Given $f,g$ two polynomials, then $ New(f*g)= New(f) + New(g) $, where  $*$  is the usual product of polynomials and the sum is the \textbf{Minkowski sum} defined for polytopes, see ref., \cite{BSturmfels1996}.
\end{lemma}

\begin{proposition}\label{Proposition 2.0.0.} Let  $ I $ be an ideal of the affine toric variety $ X_{\sigma} \, \subseteq \, \mathbb{C}^{n}$. Then define
\begin{center}
$  I(X_{\sigma})= \langle \, t^{l_{+}} - t^{l_{-}} \vert l \, \in L \, \rangle =\langle \, t^{\alpha} - t^{\beta} \vert \alpha - \beta \, \in L, \, \alpha, \, \beta \in \, \mathbb{Z}_{+}^{n} \, \rangle $,
\end{center}
where $ L $ is the kernel of the following morphism $ 0 \longrightarrow L \longrightarrow \mathbb{Z}^{n} \longrightarrow M $ and $ M $ is a monoid  such that $ M \simeq \mathbb{Z}^{d} $. The elements of $ l \, \in L $ satisfies  $ \sum_{i=1}^{n} l_{i}m_{i}=0 $.\end{proposition}

\begin{definition}\label{Definition 2.1.3.} Let be $ L \, \subseteq \, \mathbb{Z}^{n} $, a sub-lattice.\\
\textbf{(a).} The ideal $ I_{L} = \langle \,  t^{\alpha} - t^{\beta} \vert \alpha - \beta \, \in L, \, \alpha, \, \beta \in \, \mathbb{Z}_{+}^{n}  \, \rangle$, is called a \textbf{lattice ideal}.\\
\textbf{(b).} A prime lattice ideal is called a \textbf{toric ideal.}\\\end{definition}
\begin{proposition}\label{Proposition 2.1.0.} An ideal $ I \, \subseteq \, \mathbb{C}[t_{1},...,t_{n}] $ is toric if and only if it is prime and it is generated by binomials.
\end{proposition} One sees the details of the proof, ref. \cite{DCoxLittleSchenck2010}.

\subsection{Toric Morphisms and Gluing Maps.}

\begin{definition}\label{Definition 2.3.0.} Let  $ \Phi :\mathbb{C}^{k} \longrightarrow \Phi(\mathbb{C}^{k}) $ be a monomial map, i.e., each component non zero of $ \Phi $ is a monomial with coordinates in $ \mathbb{C}^{k} $, and let $ X_{\sigma} \hookrightarrow  \mathbb{C}^{k}$ and $ X_{\sigma'} \hookrightarrow  \mathbb{C}^{m}$  be inclusions of toric affine varieties. If $ \Phi (X_{\sigma}) \, \subset \, X_{\sigma'} $, then $ \varphi  := \Phi \vert _{X_{\sigma}}$ is called a \textbf{toric affine morphism} of $ X_{\sigma} $  to $ X_{\sigma'} $. If $ \varphi $ is  bijective and its  inverse map $ \varphi^{-1} :X_{\sigma'} \longrightarrow X_{\sigma} $ is also a toric morphism, then  $ \varphi $ is called an \textbf{affine toric isomorphism} and it is denoted by $ X_{\sigma} \simeq X_{\sigma'} $, \cite{GEwald1993}.
\end{definition}

\begin{proposition}\label{Proposition 2.3.0.} Every toric morphism $ \varphi : X_{\sigma} \longrightarrow X_{\sigma'}$ determines a monomial homomorphism $ \varphi^{*} : R_{\sigma'} \longrightarrow R_{\sigma}  $ and reciprocally, \cite{GEwald1993}.\end{proposition}

\begin{definition}\label{Definition 2.3.1.} For two lattice cones $ \sigma \, \subset \mathbb{R}^{n}= lin(\sigma) $ and $ \sigma' \, \subset  \mathbb{R}^{m}= lin(\sigma') $, we say that $ \sigma $ and $ \sigma' $ are isomorphic  and denote  $ \sigma \, \simeq \, \sigma'$, if $ m= n $ and  there exists an uni modular transformation $ L : \mathbb{R}^{n} \longrightarrow \mathbb{R}^{n} $ such that $ L(\sigma')=\sigma $. The monoids $ \sigma \cap \mathbb{Z}^{n} $ and $ \sigma' \cap \mathbb{Z}^{n} $ are isomorphic also.\end{definition}
\begin{definition}\label{Definition 2.3.2.} Given $ R_{\sigma} $ and $ R_{\sigma'} $ two $ \mathbb{C}-$ algebras, there is a \textbf{monomial isomorphic},    $ R_{\sigma} \simeq R_{\sigma'} $, if there exists an invertible monomial homomorphism $ R_{\sigma} \longleftrightarrow R_{\sigma'} $\end{definition}

\begin{theorem}\label{Theorem 2.3.0.} Set $ \sigma \, \subset \mathbb{R}^{n}= lin(\sigma) $ and $ \sigma' \, \subset \mathbb{R}^{m}= lin(\sigma') $, then  the following conditions are equivalent, \cite{GEwald1993}:
\begin{center}
\textbf{(a).} $ \sigma \, \simeq \, \sigma' $  {} {} {}  \textbf{(b).} $ R_{\sigma} \simeq R_{\sigma'} $  {} {} {} \textbf{(c).} $ X_{\sigma} \simeq X_{\sigma'} $.
\end{center}
\end{theorem}
\begin{proof} The implications a) $ \Rightarrow $ b)$ \Rightarrow $ c) are proven by means of the following diagram and we prove that it is commutative.
\begin{center}

$ \qquad \qquad \sigma \quad \longrightarrow \quad R_{\sigma}  \quad \hookrightarrow\quad X_{\sigma}=$\textbf{Spec}$(R_{\sigma})$\\
$ \qquad \qquad \downarrow \uparrow  L^{-1}  \qquad  $ $ \downarrow \uparrow   \psi^{-1} \qquad \quad \quad\quad\quad $ $\downarrow \uparrow \varphi^{-1} $\\
$ \qquad \qquad \sigma^{'} \quad \longrightarrow \quad R_{\sigma^{'}} \quad \hookrightarrow\quad X_{\sigma^{'}}=$\textbf{Spec}$(R_{\sigma^{'}})$
\end{center}
We define the monomial homomorphisms $ h_{\sigma}:\sigma \longrightarrow R_{\sigma}$, $ h_{\sigma^{'}}:\sigma^{'} \longrightarrow R_{\sigma^{'}}  $, $ j_{\sigma}:R_{\sigma} \longrightarrow X_{\sigma} $, $ j_{\sigma^{'}}:R_{\sigma'} \longrightarrow X_{\sigma'} $. From the hypothesis  one has that $ \sigma \backsimeq \sigma^{'} $ then there  exists a uni-modular transformation $ L $ such that $ L(\sigma)=\sigma^{'} $ and its  inverse transformation $ L^{-1}(\sigma^{'})=\sigma $ is well defined. Then one does the following monomial homomorphisms:

 \begin{center}
 $  h_{\sigma}(a)= \sum \lambda_{a}t^{a} \, \in \, R_{\sigma}$ and $ a\in \, supp(h_{\sigma}) \subset \sigma $,\\
 $ h_{\sigma^{'}}(a')= \sum \lambda_{a'}t^{a'} \, \in \, R_{\sigma^{'}} $ and $a' \in \, supp(h_{\sigma^{'}}) \subset \sigma^{'}   $,\\
 $ \psi (h_{\sigma}(a))=\sum \lambda_{a'}t^{L(a)} \, \in \, R_{\sigma^{'}}$ and $ L(a)=a' \in \, supp(\psi) \subset \sigma^{'}  $,\\
 $ \psi^{-1} (h_{\sigma^{'}}(a'))=\sum \lambda_{a}t^{L^{-1}(a')} \, \in \, R_{\sigma}$ and $ L^{-1}(a')=a \in \, supp(\psi^{-1}) \subset \sigma  $.\\
 \end{center}
Choosing the prime generators $ t^{a} \, \in \, R_{\sigma}, \, a \,\in\, \sigma$ and $ t^{a'} \, \in \, R_{\sigma^{'}}, \, a' \,\in\, \sigma^{'} $, define:
\begin{center}
$  j_{\sigma}(t^{a})=\langle t^{a} \rangle \, \in \, X_{\sigma}=$\textbf{Spec}$(R_{\sigma})$,\\
$  j_{\sigma^{'}}(t^{a'})=\langle t^{a'} \rangle \, \in \, X_{\sigma^{'}}=$\textbf{Spec}$(R_{\sigma^{'}})$,\\
$  \varphi(\langle t^{a} \rangle)=\langle t^{L(a)} \rangle \, \in \, X_{\sigma^{'}}=$\textbf{Spec}$(R_{\sigma^{'}})$,\\
$  \varphi^{-1}(\langle t^{'a} \rangle)=\langle t^{L^{-1}(a')} \rangle \, \in \, X_{\sigma}=$\textbf{Spec}$(R_{\sigma})$.
\end{center}
where  $ \langle t^{L(a)} \rangle $ and  $ \langle t^{L^{-1}(a')} \rangle $  are prime ideals like a realization, respectively, of the spectrum \textbf{Spec} of the coordinate rings $ R_{\sigma} $ and $ R_{\sigma^{'}} $.

We see easily that these monomial homomorphisms accomplish the following identities, without lost of generality,  $ \lambda_{a}=\lambda_{a'}=1 $ so: $ L\circ L^{-1}=id_{\sigma} $, $ L^{-1}\circ L=id_{\sigma^{'}} $, $ \psi \circ \psi^{-1}=id_{R_{\sigma^{'}}} $, $ \psi^{-1} \circ \psi=id_{R_{\sigma}} $, $ \varphi^{-1}\circ \varphi = id_{X_{\sigma}} $, $\varphi \circ \varphi^{-1} = id_{X_{\sigma^{'}}}  $.
The isomorphisms $ L $, $ \psi $, $ \varphi $, are isomorphisms of, cones, algebras of coordinate rings, and isomorphisms of toric varieties (toric morphism) respectively, and the first are well defined; one is a  uni-modular transformation and the second one is an isomorphisms of algebras. It only remains to proof $ \varphi $ is a toric morphism. Define the monomial homomorphism $ \Phi:\mathbb{C}^{n} \longrightarrow \mathbb{C}^{n} $ by $ \Phi(\langle t^{a} \rangle)=\langle t^{a'} \rangle\, \ni \, \Phi(X_{\sigma}) \subset X_{\sigma^{'}} $. This homomorphism induces the morphism $ \varphi $ which is bijective, so that for the generator $ t^{0}=1_{X_{\sigma}} $ as lattice vector $ a=0\, \in \, \sigma $, $ \varphi(t^{0})=t^{L(0)}=1_{X_{\sigma^{'}}} $. Then $ \varphi $ is injective and consider  the generator $ t^{a'}\, \in \, X_{\sigma^{'}} $. Since  $ L(a)=a'\Rightarrow \,\exists \,t^{a} \, \in \, X_{\sigma} \, \ni \,\varphi(t^{a})=t^{L(a)}=t^{a'} $, then $ \varphi $ is surjective. Note that   $ \varphi=\Phi\mid _{X_{\sigma}} $; in the same way one can see that $ \varphi^{-1} $ is a toric morphism too. Therefore, $ \varphi $ is a toric isomorphism. On the other hand,  $ \psi(h_{\sigma}(a))=h_{\sigma^{'}}(L(a)) $, $ \varphi(j_{\sigma}(t^{a}))=j_{\sigma^{'}}(t^{a'}) $; which proves that the   diagram commutes and one  obtains the isomorphisms wished. For  details of the  implication c) $ \Rightarrow $ a), see \cite{GEwald1993}. \textbf{q.e.d.} \end{proof}

\begin{definition}\label{Definition 2.3.3.} Recall  that a  \textbf{complex projective n-space $ \mathbb{C}P^{n} $} is the space of class of equivalence of pairs of points such that it consists of lines on $ \mathbb{C}P^{n}=\mathbb{C}^{n+1} / \sim  $. The relationship between points $ \sim $ is of the following manner, given any vector $ v:= (\eta_{0},...,\eta_{n}) $ it defines a line $ \mathbb{C}*v $ and two of said vectors $ v \sim v' \in \mathbb{C}^{n+1}\setminus \lbrace 0 \rbrace $ define the same line if and only if, one is a scalar multiple of the other.\end{definition}\
In the next example we point the important relationship of basis Hilbert and the Theorem \ref{Theorem 2.3.0.}, this connection is the great importance for the applications in the following sections; the example of Hirzebruch surface is possible consulting it in; Ewald, \cite{GEwald1993}.
\begin{example}\label{Example}. By $ H_{k} $ we mean the \textbf{Hirzebruch surface}. We consider a hyper surface in $ \mathbb{C}P^{1}\times \mathbb{C}P^{2}=\lbrace ([\eta_{0},\eta_{1}],[\zeta_{0},\zeta_{1},\zeta_{2}]) : (\eta_{0},\eta_{1}) \neq (0,0),(\zeta_{0},\zeta_{1},\zeta_{2})\neq (0,0,0) \rbrace$ determined by the equation, see example given in \cite{GEwald1993},\\
\begin{center}
$ \eta^{k}_{0}\zeta_{0}=\eta^{k}_{1}\zeta_{1},\quad k \in \mathbb{Z}. $
\end{center}
Applying Theorem \ref{Theorem 2.3.0.}, as in previous examples,  one  has the  isomorphic coordinate rings  by each one of the affine charts associated to this surface. Determining  the Newton polytopes of fan $ \Sigma $,so as its dual cones, it follows there four planes which are affine charts, and its gluing depend of $ k $, thus,\\
\begin{center}
$ R_{\sigma^{\vee}_{0}}= \mathbb{C}[z^{e_{1}},z^{e_{2}}] = \mathbb{C}[z_{1},z_{2}]$,\\
$ R_{\sigma^{\vee}_{1}}= \mathbb{C}[z^{-e_{1}},z^{e_{1}+ke_{2}}]=\mathbb{C}[z^{-1}_{2},z_{1}z^{k}_{2}]$,\\
$  R_{\sigma^{\vee}_{2}} = \mathbb{C}[z^{-e_{1}},z^{e_{2}}]=\mathbb{C}[z^{-1}_{1},z_{2}]$,\\
$ R_{\sigma^{\vee}_{3}} =\mathbb{C}[z^{-e_{1}-ke_{2}},z^{-e_{2}}]=\mathbb{C}[z^{-1}_{1}z^{-k}_{2},z^{-1}_{2}] $;\\

which implies the following toric varieties:\\

$X_{\sigma^{\vee}_{0}}=$\textbf{Spec}$ (\mathbb{C}[z_{1},z_{2}])$ ;\\
$X_{\sigma^{\vee}_{1}}=$\textbf{Spec}$ (\mathbb{C}[z^{-1}_{2},z_{1}z^{k}_{2}])$ ;\\
 $X_{\sigma^{\vee}_{2}}=$\textbf{Spec}$ (\mathbb{C}[z^{-1}_{1},z_{2}]) $;\\
 $X_{\sigma^{\vee}_{3}}=$\textbf{Spec}$ (\mathbb{C}[z^{-1}_{1}z^{-k}_{2},z^{-1}_{2}]) $.
\end{center}
Where the fan $ \Sigma^{\vee} $ formed by the cones $\sigma^{\vee}_{0}=Con(e_{1},e_{2})  $, $ \sigma^{\vee}_{1}=Con(-e_{1},e_{1}+ke_{2}) $, $ \sigma^{\vee}_{2}= Con(-e_{1},e_{2}) $, $ \sigma^{\vee}_{3}=Con(-e_{1}-ke_{2},-e_{2} )$ are the Hilbert basis $ H_{\Sigma} $ associated to the fan $ \Sigma=\lbrace \sigma_{0}, \sigma_{1}, \sigma_{2} , \sigma_{3} \rbrace  $, with $ \sigma_{0}= $Con$ (e_{1}, e_{2}) $, $ \sigma_{1}= $Con$ (-e_{2}, -ke_{1} + e_{2}) $, $ \sigma_{2}= $Con$ (-e_{2}, e_{1}) $, $ \sigma_{3} =$Con$ (-ke_{1} + e_{2}, e_{1})$. This technique will be applied in the example of three layer perceptron.\\

Following this re parametrization of polynomials one can see that it is convenient to work in complex projective spaces according to Theorem 1.\end{example}

\textbf{Gluing Maps.}\\
\begin{lemma}\label{Lemma 2.3.0.} Let $ \sigma $  be a lattice cone and set $ \tau \, \preceq \, \sigma  $. The natural identification,
\begin{center}
$  X_{\tau^{\vee}} \: \simeq \: X_{\sigma^{\vee}} \setminus \lbrace  u_{k}= 0 \rbrace$.
\end{center}
where $ u_{k} $ is the last generator of the representation of the coordinate ring associated to $ X_{\sigma^{\vee}} $, see details of the proof, ref., \cite{GEwald1993}.\end{lemma}
\begin{definition} \label{Definition 2.3.3.1.} The isomorphism,
\begin{center}
$ \psi_{\sigma, \sigma'} : \,  X_{\sigma^{\vee}} \setminus \lbrace  u_{k}= 0 \rbrace \longrightarrow  X_{\sigma'^{\vee}} \setminus \lbrace  v_{l}= 0 \rbrace $.
\end{center}
is called \textbf{gluing morphism}, which glues the varieties $ X_{\sigma^{\vee}} $ and $ X_{\sigma'^{\vee}} $ in the variety $ X_{\tau^{\vee}} $.\end{definition}
\subsection{Toric Resolution.}
\begin{definition}\label{Definition 2.4.0.} (Singularity) Let $ X_{\sum} $  be a n-dimensional toric variety and let $ \sum $ be a re\-gu\-lar fan. A point $ p \, \in \, X_{\sum} $ is called \textbf{singular} or \textbf{singularity} of $ X_{\sum} $, if $p$ belongs to an  affine chart $ X_{\sigma^{\vee}} $  where $ \sigma \, \in \, \sum $ which is not of the form $ \mathbb{C}^{k} \times (\mathbb{C}^{*})^{n-k} $. For details of the proof, see ref., \cite{GEwald1993}.\\
\end{definition}

\begin{theorem}\label{Theorem of Hironaka-Atiyah} (Hironaka-Atiyah) Let $ f $ be a real analytical function in a neighborhood  of $ \omega = (\omega_{1},....,\omega_{n})\in \mathbb{R}^{n} $ such that  $ f(\omega)= 0 $. Then there exists an open set $ V \subset\mathbb R$, a real analytical variety $ U $ and a proper analytical map $ g :U  \to V$ such that:\\
(a) $ g\quad :U-\epsilon \longrightarrow V-f^{-1}(0) $ is an isomorphism, where  $ \epsilon =g^{-1}(f(0)) $,\\
(b) For each $ u \in U $, there  exist local analytical coordinates  $ (u_{1},....,u_{n}) $ such that $ f(g(u))=\pm u^{s_{1}}_{1}u^{s_{2}}_{2}*...*u^{s_{n}}_{n} $, where $ s_{1},....,s_{n} $ are  non negative integers; see ref. \cite{SWatanabe52001}.
\end{theorem}

The previous theorem is a version of the well-known theorem of resolution of singula\-ri\-ties established by  Hironaka in algebraic geometry, see, ref. \cite{HHironaka1964}, \cite{SWatanabe22001}.\\

\begin{theorem}\label{Theorem of toric modification} Let $ X_{\sum} $ be a regular toric variety, and let $ X_{\sum_{0}} $ be a toric invariant sub variety defined by the star $ st(\sigma,\sum) \backsimeq  \sum_{0}  $ of $ \sigma $ into $ \sum $; $ 1 < k:=dim \sigma \leqslant n $.\\
(a) Under toric blow up $ \psi^{-1}_{\sigma} $, any point $ x \in X_{\sum_{0}} $ is  substituted by a k-dimensional \textbf{(k-1) projective space.}\\
(b) The blow down $ \psi_{\sigma} $ is a toric morphism which is bijective in the outside of $ \psi^{-1}_{\sigma} $.\\
See ref. for the proof of this fact, \cite{GEwald1993}.
\end{theorem}

\section{Singular Statistical Learning.}

In this section we will focus on the statistical learning machine. Given a probability space $ (\Omega, \mathbb{F}, P)  $, where $ \Omega $ is a set of events, $ \mathbb{F} $  is a $ \sigma-$algebra on $ \mathbb{F} $, and  $ P $ is a measure of Kolmogorov, one can compute  the predictive probability $ P(y \vert x,\omega) $ of the output variable $ y \in \mathbb{R}^{n}$ given $ x \in \mathbb{R}^{m}$ and a parameter vector $ \omega \, \Theta \subseteq \mathbb{R}^{n}$.  This probability is factorized according to  Bayes theorem and  to Theorem of Hammersley and Clifford; that is,   $ P(X \vert Y, \Theta) $ satisfies the local property of Markov and it can be factorized through an undirected graph $ G =(E,V)$, also it is represented by  a toric variety. \\

\begin{definition}\label{Definition 4.1.7} (Identifiable and non identifiable Machines) Let  $ (\Omega, \mathbb{F}, P)  $ be a probability space and let $ P(y \vert x,\omega) $  be the probabilistic inference or prediction probability of a statistical machine, where $ y \in \mathbb{R}^{n} $ is an     output vector  and  $ x \in \mathbb{R}^{m} $  is an input vector. If $  \omega \mapsto P(y \vert x,\omega) $ is an  injective mapping is we say the machine is \textbf{identifiable}, see  Watanabe \cite{SWatanabe42001}. If the mapping is not injective, then we say that the machine is a \textbf{ non identifiable machine}. \end{definition}

The probability densities of the learning machines are defined in the probability space $ (\Omega, \mathbb{F}, P)  $ and are denoted as follows:
\begin{itemize}
\item \textbf{prediction of the vector } $P(y \vert x, \omega)  $, {} $ y \in \, \mathbb{R}^{M}$ ,
\item \textbf{true inference of the machine} $ q(y\vert x) $,
\item  $ q(y \vert x)q(x) $ is the \textbf{ distribution of probability with which are taken and trained the set of examples} of inference machines in an independent way.
\end{itemize}

\begin{definition}\label{Definition 4.1.8} Let $ \omega_{0} \in \Theta \, \subset \,\Omega$   be a parameter such that $ P(y \vert x,\omega_{0})=q(y \vert x) $; which means, that the parameter $ q(y \vert x) $    (which establishes   the true inference of the statistical machine)  is equal  to the  predictive probability density of the output vector $ y \in \mathbb{R}^{n} $. For non identifiable machines this parameter is not unique. Even more, the set of these  parameters is called \textbf{space of true parameters}  and it is denoted  by \\
\begin{center}
$ W_{0}= \lbrace \omega_{0} \in \Theta \, \subset \, \Omega :P(y \vert x,\omega_{0})=q(y \vert x) \rbrace $.
\end{center}
It is well known that $W_{0}$ is a  sub-variety formed by singular points. If these probability densities are analytic functions, then $W_0$ is called an \textbf{analytic set}. But, if these probability densities are polynomials, then $W_0$ is called an \textbf{algebraic set}. These sets are very important for our study.
\end{definition}

\textbf{Watanabe Theorems.}\\

\begin{theorem}\label{(1) Theorem, Watanabe,} (1)Theorem, Watanabe, \cite{SWatanabe52001}, \cite{SWatanabe2005}. Suppose that $ f $ is an analytic function and $ \varphi$ is a probability density function both defined in $  \mathbb{R}^{d} $. Then, there exists a real constant $ C $ such that
\begin{center}
$ G(n) \leq \lambda_{1} \log n - (m_{1} - 1)\log (\log n) + C$,
\end{center}
 for any natural number $ n $. The rational number $ -\lambda_{1}$ {} $(\lambda_{1} > 0) $ and the natural number $ m_{1} $ are the  largest poles   of a  meromorphic function which is analytical continuation of
\begin{center}
$ J(\lambda)= \displaystyle \int_{f(\omega)< \epsilon} f(\omega)^{\lambda}\varphi'(\omega) d\omega, \quad (Re(\lambda)> 0), $
\end{center}
where $ \epsilon > 0 $ is a constant, and $ \varphi'(\omega) $ is a function of class $C^{\infty}_{0}  $ satisfying  $ 0 \leqslant \varphi'(\omega) \leqslant \varphi(\omega)$.\end{theorem}

\begin{definition}\label{Definition 4.3.1.} The poles of  the function $ J $ belong to the intersection between the negative real semi axes and the set $ \lbrace m+\nu;  \ m=0,-1,-2,...,b(\nu)=0  \rbrace $. Denoting these poles in a decreasing manner:  $ -\lambda_{1},-\lambda_{2},-\lambda_{3},...,-\lambda_{k} $, where $ \lambda_{k} $ is a  rational number, and the  multiplicity of $ -\lambda_{k} $ is denoted by $ m_{k} $.
\end{definition}

\textbf{ Condition (A)}. Let $ \psi(x,\omega) $  be  a  real valued function, where  $ (x,\omega) \in \mathbb{R}^{M} \times \mathbb{R}^{d} $, such that:\\
(1) $ \psi(x,\cdot) $ is an analytic function  on  $ W=supp(\varphi) \subset \mathbb{R}^{d} $ which can be extended to a holomorphic function on some open set $ W^{*} $, where $ W\subset W^{*} \subset \mathbb{C}^{d} $, and  $ W^{*} $ is independent of $ x \in supp(q) \subset \mathbb{R}^{M}$.\\
(2) $ \psi(\cdot,\omega) $ is a measure function on  $ \mathbb{R}^{M} $, which  satisfies:
\begin{center}
$ \displaystyle \int sup_{\omega \in W^{*}} \Vert \psi(x,\omega) \Vert^{2}q(x) dx <  \infty $,
\end{center}
where $ \Vert \bullet \Vert $ is the norm of the vector $ \psi(x,\omega) $.\\

\begin{theorem}\label{(2) Theorem, Watanabe} (2) Theorem, Watanabe \cite{SWatanabe42001}, \cite{SWatanabe52001}. Set a constant $ \sigma > 0 $. Let $ \varphi $  be a probability density of class $ C^{\infty}_{0} $.  We will consider   the statistical learning machines characterized by the following   true inference of machine:
\begin{center}
$ P(y \vert x,\omega) =\dfrac{1}{(2\pi \sigma^{2})^{N/2}} \exp \left( \dfrac{-\Vert y - \psi(x,\omega)\Vert^{2}}{2\sigma^{2}}\right) $,
\end{center}
where both $ \psi(x,\omega) $ y $ \Vert \psi(x,\omega) \Vert^{2} $ satisfies the condition (A). Then there exists a constant $ C' > 0 $ such that \begin{center}
$ \vert G(n) - \lambda_{1} \log n + (m_{1} - 1)\log \log n \vert \leqslant C'$,
\end{center}
for any natural number $ n $,  where the rational number $ -\lambda_{1} (\lambda_{1} > 0) $ and a natural number $ m_{1} $ are the  largest poles   of a  meromorphic function which is analytical continuation of
\begin{center}
$\displaystyle  J(\lambda)= \int _{f(\omega) < \epsilon} f(\omega)^{\lambda} \varphi(\omega) d\omega, (Re (\lambda)> 0) $,
\end{center}
where $ \epsilon > 0 $ is a constant.\end{theorem}

\textbf{Learning Curves and Resolution of Singularities.}\\
It is well known that  the  regular statistical models in which  $ \lambda_{1}=d/2 $ and $ m_{1}=1 $ are special cases of Theorem (2) (Watanabe). In models of non identifiable machines, generally, the  bayesian neural networks have different values of $ \lambda_{1} \leq d $  and  $ m_{1} \geq 1 $.  It is work of the algebraic geometry to find the poles $ \lambda_{1} $ and $ m_{1} $, of the meromorphic function $ J(\lambda) $ defined in theorem (1) and (2) (Watanabe),  by means of   techniques of resolution of singularities suggested by Watanabe, \cite{SWatanabe42001}, \cite{SWatanabe52001}, \cite{SWatanabe62001}, such as toric modification and blow up, in the algebraic set $ \lbrace \lambda \in W : H(\lambda)=J(\lambda) =0 \rbrace. $ .\\

\begin{corollary}\label{Corollary 4.3.1} Suppose the hypothesis of Theorem  \ref{(2) Theorem, Watanabe}. If $ c(n+1) - c(n)= o\left( \dfrac{1}{n\log n}\right)  $, then the \textbf{learning curve} is given by,
\begin{center}
$ K(n)= \dfrac{\lambda_{1}}{n} + \dfrac{m_{1} - 1}{n \log n} + o\left( \dfrac{1}{n\log n}\right)  $.
\end{center}
Using this formula in regular models one has that  $ \lambda_{1} =d/2$ and $ m_{1}=1 $. For non identifiable models, such as bayesian neural networks, the corresponding values are  $ \lambda_{1} \leq d/2 $ y $ m_{1} \geq 1 $.\\
\end{corollary}

\begin{corollary}
\label{Corollary 4.3.2.} Suppose the hypothesis of Theorem \ref{(1) Theorem, Watanabe,}. If  $ \varphi'(\omega) > 0 $ for each $ \omega_{0} \in W_{0} $,  then $ \lambda_{1} \leq d/2 $ where $ d $ is the \textbf{dimension of parameter spaces}. See ref. \cite{SWatanabe42001}, \cite{SWatanabe52001}.\end{corollary}

\begin{definition}\label{Definition 4.1.9} The \textbf{Kullback distance}, or  \textbf{information entropy}, of a statistical machine
quantifies  the distance between the predictive probability $ P(y \vert x,\omega) $, of the output variable $ y \in \mathbb{R}^{N}$, and the true statistical inference of the machine $q(y \vert x)  $. \\
\begin{center}
\textbf{(Kullback distance)} $\displaystyle H(\omega) = \int \log \dfrac{q(y \vert x)}{P(y \vert x,\omega)}q(y \vert x)q(x) dxdy$.
\end{center}
where $ q(x) $ is the true probability of the input variable $ x $.\end{definition}

The Kullback distance induces other important definitions.

\begin{definition}\label{Definition 4.1.10.}  The \textbf{learning curve} of a statistical machine or \textbf{generalization of the error}, Watanabe \cite{SWatanabe32001}, is given by\\
\begin{center}
$ \displaystyle K(n)= E_{n} \left\lbrace  \int \log \dfrac{q(y \vert x)}{P_{n}(y \vert x,\omega)}q(y \vert x)q(x) dxdy \right\rbrace  $,
\end{center}
where $ E_{n} \lbrace \bullet \rbrace $ is the expected value over all pairs of trained examples  by the machine, and $ P_{n}(y \vert x,\omega) $ is the mean density probability over all posterior probabilities of the output of the machine.
\end{definition}

\textbf{Algebraic geometry of Statistical machines.}\\

\begin{definition}\label{Definition 4.1.11} It is important to comment that an \textbf{algebraic set} $ W_{0} $ is, equivalently, defined by
\begin{center}
$ W_{0}= \lbrace H(\omega)=0 : \omega \in \Theta \rbrace$.
\end{center}
This set is not empty and is the principal set of our  study related with   singular machines.
\end{definition}

\section{Singular Machines.}

\begin{definition}\label{Definition 4.2.1} Let  $(\Omega, \mathbb{F}, P) $ be  a probability space and let  $ y \in \mathbb{R}^{N}$  be a random vector. Define the \textbf{Fisher information matrix}, as follows, see A.S. Poznyak, \cite{ASPoznyak2009}:

\begin{center}
$\displaystyle  I(\omega) = E_{n} \left\lbrace \nabla_{\omega} \log P_{n}(y \vert x,\omega)   \nabla_{\omega}^{\intercal} \log  P_{n}(y \vert x,\omega) \right\rbrace = \, \, \int_{\omega \in \mathbb{R}^{M}} \left\lbrace \nabla_{\omega} \log P_{n}(y \vert x,\omega)   \nabla_{\omega}^{\intercal} \log  P_{n}(y \vert x,\omega) \right\rbrace P_{n}(y \vert x,\omega)q(x) dxdy $,
\end{center}
where $ P_{n} $ is given  in Definition \ref{Definition 4.1.10.} In general terms, this expression can be understood as a metric in the parameter space whenever the matrix is  positive defined.
\end{definition}

\begin{definition}\label{Definition 4.2.2.} A statistical learning machine is called \textbf{regular learning machine} if the Fisher information matrix is positive defined, otherwise, it is called  a \textbf{singular learning machine} if there  exists a parameter $ \omega \in \Theta $ (called singularity of the Fisher information matrix) such that \textbf{det}$ I(\omega)=0 $. These singularities are several  and the probability of the parameter $ \omega $ can not be approximated by a quadratic form in the sense of differential geometry, see the regular statistical machines, see ref. \cite{SWatanabe42001}.\end{definition}

\subsection{Effect of the Singularities in the Statistical Learning.}

In the following we define the mean empirical Kullback distance as:
\begin{center}
$ H_{n} = \dfrac{1}{n} \sum_{i=1}^{n} \log \dfrac{q(y_{i} \vert x_{i})}{P(y_{i} \vert x_{i},\omega)}$,
\end{center}
and let $ H(\omega) $  be an usual Kullback distance as we previously saw, if there exists a parameter $ \omega_{0} $ such that $ H(\omega_{0})=0 $, then $ H(\omega) $ satisfies the statement of Theorem \ref{Theorem of Hironaka-Atiyah}. Therefore,  there exists a variety $ U $ and  a resolution map $ g: U \mapsto W $, such that,
\begin{center}
$ H(g(u))=A(u)^{2} $ with $ A(u)= u^{k_{1}}_{1}*...*u^{k_{d}}_{d} $,
\end{center}
and the empirical distance can be written as above; for more details see ref. \cite{SWatanabe32001}, \cite{SWatanabe42001}, \cite{SWatanabe62001}.

\begin{definition}\label{Definition (Synaptic Function)} According to the previous notations, the synaptic function of a statistical learning machine is given as follows:
 \begin{center}
$ \psi(x,y,u)=\dfrac{1}{A(u)}\left( H(g(u)) - \log\dfrac{q(y \vert x)}{P(y \vert x,g(u))}\right)  $.
\end{center}
 The function $ \psi(x,y,u) $ can be written as $ \psi(x,y,g^{-1}(u)) $ if $ H(\omega)\neq 0 $. However, it is well defined, in general, when  $ H(\omega)=0 $. On the other hand, we proved that $ \psi(x,y,u) $ is an analytic function of $ u $, whenever $ H(g(u))=0 $. From the property of normal crosses of $ A(u) $, one can see that   $ \psi(x,y,u) $ is well defined in the variable $ u $, see ref., \cite{SWatanabe42001}, \cite{SWatanabe52001}.\end{definition}
 \textbf{Learning Coefficient.} In this part of our work, we compute the learning coefficient of the following statistical learning machine:\\
\begin{center}
$P(y \vert x,a,b) = \dfrac{1}{\sqrt{2\pi}} \exp(-\dfrac{1}{2}(y - af(b,x))^{2})$.
\end{center}
The true statistical inference of the machine is given by:
\begin{center}
$ q(y \vert x) =\dfrac{1}{\sqrt{2\pi}} \exp\left( \dfrac{-1}{2}(y -\dfrac{a_{0}f(b_{0},x)}{\sqrt{n}})^{2}\right) $,
\end{center}
where
\begin{center}
$ \int \dfrac{\psi(b)db}{\Vert f(b) \Vert} < \infty $.
\end{center}
Then, the learning curve of this machine can be expanded asymptotically by:
\begin{center}
$ G(n)=\dfrac{\lambda(a_{0},b_{0})}{n} + o\left( \dfrac{1}{n}\right) $.
\end{center}
The \textbf{learning coefficient}, which is independent of $ n $,  is given  by:
\begin{center}
$ \lambda(a_{0},b_{0})=\dfrac{1}{2} \left\lbrace  1 + a^{2}_{0}\Vert f(b_{0})\Vert^{2} - \sum_{j=1}^{J}a_{0}f_{j}(b_{0})E_{g}\left[ \dfrac{\partial}{\partial g_{j}} \log Z(g)\right]  \right\rbrace $,
\end{center}
 see, Watanabe, \cite{SWatanabe42001}, where $ g=\lbrace g_{j} \rbrace$ is a random variable  subject to the dimensional gaussian distribution $ J $,  whose mean is zero and its covariance matrix is the identity. By  $ E_{g} $ we mean  the expecting values on $ g $, and set
$(g) = \int \exp(L(g))\dfrac{\psi(b)db}{\Vert f(b) \Vert},$
with $  L(b)=\dfrac{m((g + a_{0}f(b_{0}))*f(b))^{2}}{2\Vert f(b) \Vert^{2}} .$

Now, considering,  at the beginning of our example, the synaptic function $ f(b,x) $ in terms of its expansion in orthonormal basis $ e_{j} $, one has that
\begin{center}
$ P(y \vert x,a,b)=\dfrac{1}{\sqrt{2\pi}} \exp \left( \dfrac{-1}{2}(y - \sum_{j=1}^{N} ab_{j}e_{j}(x))^{2}\right).  $
\end{center}
In this case for the synaptic function $ \Psi $. When $ N \geqslant 2 $, in a model of true regression, see ref. \cite{SWatanabe42001}, this machine is singular with parameter space $ W_{0}=\lbrace (a,b), a=0, b=0, : a \in \mathbb{R}, b \in \mathbb{R}^{N} \rbrace$. Using the toric resolution, it is  parametrized by means $ \omega_{i}=ab_{i} $ and substituting in the model one has:
\begin{center}
$ P(y \vert x,a,b)=\dfrac{1}{\sqrt{2\pi}} \exp\left( \dfrac{-1}{2}(y - \sum_{j=1}^{N} \omega_{j}e_{j}(x))^{2}\right)  $.
\end{center}
With this resolution in the parameter space,  the model has become a regular model, with learning coefficient given by $ \lambda(\omega_{0}) = N/2$ for an arbitrary parameter $ \omega_{0} $. Meanwhile, without this toric resolution the learning coefficient would be given by the above expression  $ \lambda( a_{0} ,b_{0}) $. Clearly, $ \lambda(a_{0},b_{0}) \neq \lambda(\omega_{0})  $. With the previous facts, one see that  the \textbf{singularities into the parameter space play an important role in the statistical learning, \cite{SWatanabe22001}, \cite{SWatanabe32001}.}

\begin{theorem}\label{Corollary 4.3.3} Let  $P(y\vert x, \omega)  $ be   a non singular  statistical learning machine, see Definition \ref{Definition 4.2.2.}, in the probability space $ (\Omega, \mathbb{F}, P)  $, and  consider the  Kullback distance associated with its parameters space; i.e.,   $ \lbrace \lambda \in W \, \subset \, \Omega: H(\lambda)=J(\lambda) =0 \rbrace $. Then,  the following polynomial is a parametrization such that for  each  $ \Theta \, \subset \, \Omega$, there exists $ \, \omega \,\in \, \Theta \subset \Omega \subset \mathbb{C}^{n},\,   $ as we have seen, $ H(\omega)=\sum_{i=1}^{n} c_{i}\omega^{a} $  with $ c_{i} \,\in \, \mathbb{R} ,\, \omega^{a} \, \in \, \mathbb{C^{*}}$, {} for each $i$, and  $ a \, \in \, \mathbb{Z}^{n} $ being the lattice vector (see Definition \ref{Definition 2.1.2.}, Newton polytope) if and only if the lattice cone $ \sigma \subseteq \, supp(H(\omega)) $, generated by the lattice vector of support  of $ H(\omega) $, is not singular; i.e.,
\textbf{Det} $ (\sigma)=1 $.\end{theorem}
\begin{proof}  The need is a consequence of the following facts. As  $ H(\omega) $ is a re parametrization of singular polynomial $ H(\lambda) $ then  there exists  a resolution map $ g: H(\lambda) \, \longrightarrow \, H(\omega) $, by the Theorem \ref{Theorem of Hironaka-Atiyah}, such that $ H(\omega) $ is not singular. Now, let $ \sigma' \subset \, supp(H(\lambda)) $ and let  $ \sigma \subset \, supp(H(\omega)) $ be  lattice cones generated by the lattice vectors of support of $ H(\lambda) $ and $ H(\omega) $, respectively. Then, we affirm that $ g $ induces  a morphism of monomial generated finitely $ \mathbb{C}-$ algebras: $ R_{\sigma'}= \lbrace f \vert \, supp(f)\subset \, \sigma' \rbrace $ and $ R_{\sigma}= \lbrace f \vert \, supp(f)\subset \, \sigma \rbrace  $,  such that $ H(\lambda)\,\in \, R_{\sigma'} $ and $ H(\omega) \, \in \, R_{\sigma}$. The proof of this  follows from its definition as resolution map. Then Theorem \ref{Theorem 2.3.0.}, implies that  $ R_{\sigma'} \simeq R_{\sigma} \, \Longrightarrow \, \sigma' \simeq \sigma$. Therefore there exists  an uni modular transformation $ L \, \in \, \mathbb{Z}^{n} \times \mathbb{Z}^{n} $ with a matrix associated to the Hilbert basis $ H_{(\sigma')^{\vee}} $, with dual cone $ (\sigma')^{\vee} \, \Longrightarrow \, L(\sigma')=\sigma$ and with uni modularity of this transformation. Then \textbf{Det} $L(\sigma')=$ \textbf{Det} $ \sigma=1 \, \Longrightarrow \,\sigma$ is not singular.\\
 Reciprocally, let $ \sigma \subset \, supp(H(\omega)) \, \ni $ \textbf{Det} $ \sigma =1 \, \Longrightarrow \,\exists \, L \in \, \mathbb{Z}^{n} \times \mathbb{Z}^{n} \, \ni \, L(\sigma')=\sigma $ be a lattice cone  with $ L $ being an uni modular transformation, for some cone $ \sigma' \subset \, \mathbb{Z}^{n} \, \Longrightarrow \, \sigma'\simeq \sigma$. Then by Theorem \ref{Theorem 2.3.0.}, one has that this isomorphism lifts to a toric morphism $ \psi $ such that $ R_{\sigma'}\simeq R_{\sigma} $ to make it compatible with the previous notation, we do $ g=\psi $. Since the morphisms of monomial finitely generated $ \mathbb{C}-$ algebras are torics as it has been said before, then $ g $ is a resolution map of $ H(\omega) $ and therefore  this polynomial is not singular. We will prove that the Fisher information matrix $ I(\omega) $ associated to the model is not singular, i.e., \textbf{Det} $ I(\omega)\neq 0 $.\\
\textbf{Proof of the affirmation.} If $ H(\omega) $ is singular, then  the inference machine  $ P(y \vert x,\omega) $ is a model non iden\-ti\-fia\-ble, by Definition \ref{Definition 4.1.8}. Therefore, $ P(y \vert x,\omega)=q(y\vert x) $ where $ q(y \vert x) $ is the true statistical inference of the model. Applying the operator $ \nabla_{\omega} $ to  $P(y \vert x,\omega)  $, from Definition of Fisher information matrix (Definition \ref{Definition 4.2.1}), one has that  $\nabla_{\omega} P(y \vert x,\omega)= \nabla_{\omega} q(y\vert x) = 0  \,\Longrightarrow$ \textbf{Det}
$ I(\omega)=0 $, which concludes the proof, \textbf{q.e.d.} \end{proof}

\section{Applications in singular machines.}

We present an application to the learning curve in the following singular machine.\\
\textbf{Application A.} Consider  the polynomial which represents the learning curve of a perceptron of two layers $ H(a,b,c)= a^{2}b^{2}+ 2abc + c^{2} +  3a^{2}b^{4}, \ (a,b,c) \in \mathbb{R}^{3} $ which is singular  in its parameter space, $ (0,0,0) \in \mathbb{R}^{3} $. Establishing the Newton polytope,   defined by $ supp(H) $, one has the lattice cone: $ \sigma=Con ((2,2,0),(1,1,1),(0,0,2),(2,4,0)) $.\\
Now,  we get the following associated dual cone, see Theorem \ref{Theorem 2.3.0.}:
\begin{center}
$ \sigma^{\vee}= Con(2e_{1} - e_{2} - e_{3},-e_{1} + e_{2}, e_{3}) $,
\end{center}
 which gives the  Hilbert basis associated to monoid $ \sigma \cap \mathbb{Z}^{3}$.
\begin{center}
$ \mathrm{H}_{\sigma^{\vee}} = \lbrace e_{1} + e_{2},e_{1} + e_{2} + e_{3}, e_{1} + 2e_{2} \rbrace $.
\end{center}
From Theorem \ref{Theorem 2.3.0.}, one obtains  the geometric realization of the affine toric variety $ X_{\sigma'^{\vee}} $,\\
\begin{center}
$X_{\sigma'^{\vee}} $ = \textbf{Spec}$(\mathbb{C}[S_{\sigma'^{\vee}} \cap \mathbb{Z}^{3}])= \textbf{Spec}(\mathbb{C}[u_{1}u_{2},u_{1}u_{2}u_{3},u_{1}u^{2}_{2}])$.
\end{center}

One chooses the set of generators of $ X_{\sigma'^{\vee}} $, which forms a uni modular matrix $ A=Columns((1,1,0)',(1,1,1)',(1,2,0)') $, and   parametrize this system by means of monomials of Laurent.\\

By  Theorem \ref{Theorem of Hironaka-Atiyah}, one  obtains  the resolution map,
\begin{center}
$ g_{1}: (a,b,c) \longrightarrow (u_{1}u_{2},u_{1}u_{2}u_{3},u_{1}u^{2}_{2})/(0,0,0) $,
\end{center}
such that,\\
 \begin{center}
$ H(g_{1}(u_{1},u_{2},u_{3}))= u^{4}_{1}u^{4}_{2}u^{2}_{3} + 2u^{3}_{1}u^{4}_{2}u_{3} + u^{2}_{1}u^{4}_{2} + 3u^{6}_{1}u^{6}_{2}u^{4}_{3}$\\
$\qquad \quad\quad\quad\quad\quad=u^{2}_{1}u^{4}_{2}(u^{2}_{1}u^{2}_{3} + 2u_{1}u_{3} + 1 + 3u^{4}_{1}u^{2}_{2}u^{4}_{3}) $\\

$\quad\quad\quad\quad\quad=u^{2}_{1}u^{4}_{2}((u_{1}u_{3} + 1)^{2} + 3u^{4}_{1}u^{4}_{3}u^{2}_{2}) $\\

$\qquad =c^{2}_{1}((b_{1} + 1)^{2} + 3b^{4}_{1}d^{2}_{1}) $\\

$ \quad\quad=c^{2}_{1}(b'^{2}_{1} + 3(b'_{1} - 1)^{4}d^{2}_{1}) $\\

$=c^{2}_{1}(b'^{2}_{1} + 3e^{4}_{1}d^{2}_{1}) $.

\end{center}
By applying a second time the technique of resolution by means of Hilbert basis to the polynomial defined by $ h(b'_{1},d_{1},e_{1})=b'^{2}_{1} + 3e^{4}_{1}d^{2}_{1} $,  where  the basis Hilbert associated to Newton polytope of $ supp(h) $, in this manner, one has  that $ \mathrm{H_{\sigma^{\vee}}}=\lbrace e_{1},e_{1} + e_{2},e_{1} + 2e_{2} \rbrace $, and as consequence of the resolution map,
\begin{center}
$g_{2}:(b'_{1},d_{1},e_{1}) \longrightarrow (s_{1},s_{1}s_{2},s_{1}s^{2}_{2})/(0,0,0)$.
\end{center}
Then, one obtains the affine toric variety; $ X_{\sigma^{\vee}}= Spec\mathbb{C}[s_{1},s_{1}s_{2},s_{1}s^{2}_{2}] $, and so,
\begin{center}
$ H(g_{1}(g_{2}(s_{1},s_{2},s_{3})))=c^{2}_{1}s^{2}_{1}(1+ 3s^{4}_{1}s^{8}_{2}s^{2}_{2}) $\\
 $ =c^{2}_{1}s^{2}_{1}(1 + 3s^{4}_{1}s^{10}_{2}) $,
\end{center}
which is not singular in $ (0,0,0) \in \mathbb{R}^{3} $. This same fact is proved in, S. Watanabe, see, ref. \cite{SWatanabe42001}, \cite{SWatanabe62001}.\\

\textbf{Application (B). Mix of binomial distributions.} This kind of statistical learning is used by the spectral analysis of mutations, see \cite{MAoyaguiWatanabe2005}, and the  statistical machine is cha\-rac\-te\-ri\-zed by the following probabilities: \\

True probability of $ x $, $ q(x=k)=Bin_{N} (x;p^{*})= \binom {N}{x}p^{*x} (1 - p^{*})^{N - x}$.\\

True probabilistic inference of model $ P(x=k \vert w)= aBin_{N}(x,p_{1}) + (1 - a) Bin_{N}(x,p_{2}) $.\\

Parameter space $ w$ is defined by:
\begin{center}
$ w = (\lbrace  a_{i}  \rbrace^{K}_{i=1}, \lbrace p_{i} \rbrace^{K+1}_{i=1}) $,
\end{center}
where  coordinates of the parameters $ p_{i} $ are defined in the range $ 0 < p_{i} <1/2 $ and,
\begin{center}
$ a_{K+1}= 1 - \sum^{K}_{i=1} a_{i} $.
\end{center}
There holds  the following theorem proved by Watanabe, see ref., \cite{KYamazakiWatanabe2004}, immediately, the same result is proven but using Hilbert basis and toric morphism: \\

\begin{theorem}\label{Theorem} Consider a learning machine characterized by the probabilities defined above, then for a number large enough $ n $ of training examples, in accordance with Corollary \ref{Corollary 4.3.1}, its learning curve is given by:
\begin{center}
$ K(n)= \dfrac{3}{4} \log(n) + C $,
\end{center}
where $C$ is independent of $ n $.\\
\end{theorem}
\textbf{Proof.} The  Kullback information distance is given by:

\begin{center}
$ \mathrm{H}(x,a,b_{1},b_{2})= \sum^{N}_{x=0} q(x) log\left( \dfrac{q(x)}{P(x \vert \omega)} \right) $
$ =(ap_{1} + (1 - a)p_{2})^{2} + (ap^{2}_{1} + (1 - a)p^{2}_{2})^{2} +....+$\\
$ =b^{2}_{2} + (ab^{2}_{1} + (b_{2} - ab_{1})^{2})^{2} +...+$ major order terms.
\end{center}

which is singular in $ (0,0,0) \in \mathbb{R}^{3}$. According to Theorem  \ref{Theorem of Hironaka-Atiyah}  and Hilbert basis lemma, and to our technique with toric morphisms, one can see that the previous  polynomial is generated by the ideal  $ \mathrm{I}< \mathbb{C}[a,b_{1},b_{2}] $\\
\begin{center}
$ \mathrm{I}:= < b_{2}^{2}; ab_{1}^{2}; ab_{1}b_{2}> $
\end{center}
and the lattice cone
\begin{center}
$ \sigma = Con((0,0,2);(1,2,0);(1,1,1)). $
\end{center}
Computing  the  geometric realization of the affine toric variety $ X_{\sigma^{\vee}} $, associated with monoid $ S_{\sigma}= \sigma^{\vee} \cap \mathbb{Z}^{3} $ and with Hilbert basis,  one obtains \\
\begin{center}
$ \mathrm{H}_{\sigma^{\vee}}= \lbrace e_{3}; e_{1} + e_{2} + e_{3}; e_{1} + 2e_{2}  \rbrace $,
\end{center}
where the toric variety is:
\begin{center}
$ X_{\sigma^{\vee}} =$\textbf{Spec}$\mathbb{C}[w_{3},w_{1}w_{2}w_{3},w_{1}w_{2}^{2}] $,
\end{center}
and the coordinate system:
\begin{center}
$a= w_{3} ; $\\
$ b_{1}=w_{1}w_{2}w_{3}; $\\
$ b_{2}=w_{1}w_{2}^{2}$,
\end{center}
and  using  this parametrization we get the resolution map $ g : X_{\sum'} \longrightarrow X_{\sum} $  such that  $ \mathrm{H}(g(w)), w=(w_{1},w_{2},w_{3}) \in \mathbb{R}^{3}$ is not singular in $ (0,0,0) $ and from Theorem \ref{Theorem of Hironaka-Atiyah};

$ \mathrm{H}(g(w))= w_{1}^{2}w_{2}^{4} + (w_{3}w_{1}^{2}w_{2}^{2}w_{3}^{2}+ (w_{1}w_{2}^{2} - w_{1}w_{2}w_{3}^{2})^{2})^{2} +...+$ order major terms),\\
$=w_{1}^{2}w_{2}^{4} + [w_{1}^{2}w_{2}^{2} + (w_{3}^{3}+(w_{2} -w_{3}^{2})^{2})]^{2} +...+$ order major terms,\\
$ =w_{1}^{2}w_{2}^{4} + w_{1}^{4}w_{2}^{4}[w_{3}^{3}(w_{2}-w_{3}^{2})^{2}]^{2} +...+$ order major terms.\\
$ =w_{1}^{2}w_{2}^{4}(1 + w_{1}^{2}(w_{3}^{6} +2w_{3}^{3}(w_{2}- w_{3}^{2})^{2}+(w_{2}-w_{3})^{4} +...+$ order major terms).

It is easy to see that writing the terms of integration  of $ J(z) $,  we get,
\begin{center}
$ \displaystyle J(z)=\int H(g(w))^{z} \vert g'(w) \vert du$
$  \displaystyle  =\int ((1 + w_{3}^{2}w_{1}^{2}+...)w_{2}^{4}w_{1}^{2})^{z} \vert w_{2}^{2}w_{1} \vert dw_{1}dw_{2}dw_{3} $
$ =\dfrac{f(z)}{4z + 3} $,
\end{center}
where the most large pole of $ J(z) $ is $ \lambda_{1}=\dfrac{3}{4} $ and multiplicity $ m_{1}=1 $, then the learning curve is given by: \\
\begin{center}
$ K(n)=\dfrac{3}{4} \log(n) + C $;
\end{center}
\textbf{q.e.d.} see ref. \cite{KYamazakiWatanabe2004}.\\

\textbf{Application (C).} The following application is the toric resolution in a perceptron of three layer on the learning curve of the same, for details of the computing of this learning curve, see, Watanabe \cite{SWatanabe42001}. We define the machine in the space probability $ (\Omega, \mathbb{F}, P) $ as follows: \\
\begin{enumerate}
\item \textit{A priori} probability distribution, $ \varphi(\omega) > 0 $.
\item Predictive probability of the vector $ y \in \mathbb{R}^{N} $,
\begin{center}
$P(y \vert x,\omega)=\dfrac{1}{(2\pi s^{2})^{N/2}}\exp \left( \dfrac{-1}{2s^{2}} \Vert y - f_{k}(x,\omega)\Vert^{2} \right)$,
\end{center}
with $ x \in \mathbb{R}^{M} $ and $ s > 0 $ is the standard deviation.
\item True probability distribution of model,
\begin{center}
$ q(y \vert x)q(x) = \dfrac{1}{(2\pi s^{2})^{N/2}}\exp \left( -\dfrac{1}{2s^{2}}\Vert y \Vert^{2}\right)  q(x)$.
\end{center}
\end{enumerate}
First we compute the Kullback distance of this machine is defined as follows,
\begin{center}
$ \displaystyle  H(a,b,c)= \dfrac{1}{2s^{2}}\int \Vert f_{K}(x,a,b,c) \Vert^{2} q(x) dx  $
 $ =\sum^{N}_{p=1} \sum^{K}_{h,k=1} B_{hk}(b,c) a_{hp} b_{kp} $.
\end{center}
with parameter space associated, and function of the hidden units given by, $ f_{K}(x,\omega)= \sum^{K}_{k=1} a_{k} \sigma(b_{k} x + c_{k}) $ {}  :
\begin{center}
$ a=\lbrace a_{k} \in \mathbb{R}^{N}; k=1,2...,K   \rbrace $\\
$ b= \lbrace b_{k} \in \mathbb{R}^{M}; k=1,2,...,K \rbrace $\\
$ c=\lbrace c_{k} \in \mathbb{R}; k=1,2,...,K \rbrace $,\\
$ a_{k}=\lbrace a_{kp} \in \mathbb{R}; p=1,2...,N    \rbrace $\\
$ b_{k}=\lbrace b_{kp} \in \mathbb{R};q=1,2,...,M \rbrace $.
\end{center}
defining $ \displaystyle   B_{hk}(b,c)=\dfrac{1}{2s^{2}}\int \sigma(b_{h}*x + c_{h})\sigma(b_{k}*x + c_{k})q(x)dx $, where $ \sigma(x)=tanh(x) $ is a synaptic function, see for details of the formulation, Watanabe \cite{SWatanabe42001}. Thus developing terms.
\begin{center}
$  H(a,b,c)=\sum_{p=1}^{N}(B_{11}(b,c)a_{1p}a_{1p} + B_{22}(b,c)a_{2p}a_{2p}+...+B_{KK}a_{Kp}a_{Kp})$\\
$ =B_{11}a_{11}^{2}+B_{22}a_{21}^{2}+....+B_{KK}a_{K1}^{2}$\\
$+B_{11}a_{12}^{2}+B_{22}a_{22}^{2}+....+B_{KK}a_{K2}^{2}  $\\
$ +B_{11}a_{13}^{2}+B_{22}a_{23}^{2}+....+B_{KK}a_{K3}^{2} $\\
$ +B_{11}a_{14}^{2}+B_{22}a_{24}^{2}+...+B_{KK}a_{K4}^{2}+ $\\
$ .............+......+......$\\
$ +B_{11}a_{1N}^{2}+B_{22}a_{2N}^{2}+....+B_{KK}a_{KN}^{2}. $
\end{center}
This polynomial seen in the coordinates $ a_{hk}\in\mathbb{R} $, is singular in $ (0,0,...,0)\in\mathbb{R} ^{N} $, we construct a toric resolution in this coordinates utilizing the concept of projective sets as previously we have seen. We define the affine charts utilizing the following projective set,
\begin{center}
$ U_{j}=\lbrace[a_{11},...,a_{1N},a_{21},...,a_{2N},...,a_{K1},...,a_{KN}]\in \mathbb{R} P^{KN-1}:a_{jj}\neq 0\rbrace$.
\end{center}
Where $ \mathbb{R} P^{KN-1}$ is the real projective space $ KN-1 $-dimensional, and also there exists a bijection as we have seen, the affine real space $ KN-1 $-dimensional  $ \mathbb{R}^{KN-1} $, given by,
\begin{center}
$ U_{j}:\mathbb{R} P^{KN-1}\longmapsto \mathbb{R} ^{KN-1}   $
\end{center}
\begin{center}
$  U_{j}:[a_{11},...,a_{1N},a_{21},...,a_{2N},...,a_{K1},...,a_{KN}]\longmapsto
(1,a_{12}a_{11}^{-1},...,a_{1N}a_{11}^{-1},a_{21}a_{11}^{-1},...,a_{2N}a_{11}^{-1},...
,a_{K1}a_{11}^{-1},...,a_{KN}a_{11}^{-1})$
\end{center}
now in these projective coordinates we redefine $ H(a,b,c)=u_{11}^{2}H_{1}(a,b,c) $, since:
\begin{center}
$ H(a,b,c)=a_{12}^{2}u_{11}^{2}B_{11}+a_{13}^{2}u_{11}^{2}B_{11}+...+a_{1N}^{2}u_{11}^{2}B_{11}+a_{21}^{2}u_{11}^{2}B_{22} +...+ a_{2N}^{2}u_{11}^{2}B_{22}+...+
a_{k1}^{2}u_{11}^{2}B_{KK}+...+a_{KN}^{2}u_{11}^{2}B_{KK}$
\end{center}
where the new coordinates in the affine space $ \mathbb{R} ^{KN-1} $ son $ (u_{11},a_{12},...,a_{1N},...,a_{K1},...,a_{KN})\\
\in \mathbb{R} ^{KN-1} $, in this new coordinate ring  we construct the lattice cone of Newton polytope associated in the re parametrized polynomial, which give us as: $ \sigma=Con(2e_{1}+2e_{2},...,2e_{1}+2e_{1N},...,
2e_{1}+2e_{K1},...,2e_{1}+2e_{KN}) $, that in matrix way give rises the following array associated to the cone,
\\
\\
\\\\
$ A_{\sigma}= $
\begin{center}
\begin{tabular}{c c c c c c c}
2&2&0&.&.&.&0 \\
2&0&2&.&.&.&0\\
2&0&0&2&.&.&0\\
.&.&.&.&.&.&.\\
.&0&.&.&.&2&0\\
2&0&.&.&.&0&2\\\\\\
\end{tabular}
\end{center}

It possible to show in a inductive way, and using of Singular program, \cite{DGPS}, that the Hilbert basis associated to this lattice cone, are given by the following matrix array,\\\\

$ H_{\sigma^{\vee}}= $
\begin{center}
\begin{tabular}{c c c c c c c}
1&0&0&.&.&.&1 \\

1&0&.&.&.&1&0\\

1&0&.&.&1&0&0\\

.&.&.&.&.&.&.\\
.&.&1&0&.&.&0\\
1&1&0&.&.&.&0\\\\

\end{tabular}
\end{center}
where the lattice vectors of this array represent us a regular lattice cone; and by The Theorem \ref{Theorem 2.3.0.}  and Theorem \ref{Theorem of toric modification}, we have a toric blow up or toric resolution, also we obtain the respectively toric variety $ X_{\sigma^{\vee}} $, taking as exponents the elements of this base for the constructing the monomial homomorphisms ( Theorem \ref{Theorem 2.3.0.} ), and so the following transformation of monomial coordinates:
\begin{center}
$ a_{11}=v_{11}; $\\
$  u_{11}=v_{11}^{-1}$,\\
$ a_{hp}=u_{11}*u_{hp}$; $ \forall $  $h\neq 1 $ \'{o} $p\neq 1$.

\end{center}
which is the re parametrization shown in, Watanabe \cite{SWatanabe42001}. Furthermore we set a extra coordinate $ u_{11} $, being that we work with the projective set $ U_{j} $, where we construct from the affine chart;
\begin{center}
$ A_{0}=\lbrace(1,a_{12}a_{11}^{-1},...,a_{1N}a_{11}^{-1},a_{21}a_{11}^{-1},...,a_{2N}a_{11}^{-1},...
,a_{K1}a_{11}^{-1},...,a_{KN}a_{11}^{-1}) \vert a_{11} \neq 0 \in\mathbb{R}^{KN-1} \rbrace $,
\end{center}
so explicitly we have the toric variety as the affine algebraic scheme
\begin{center}
 $ X_{\sigma^{\vee}} = $\textbf{Spec}$(\mathbb{C}[\sigma^{\vee} \cap \mathbb{Z}^{KN-1}])=A_{0}  $;
\end{center}
is enough to realize the toric resolution in this chart being that the toric morphism are proper, and they extend at all the variety. Finally by the Watanabe's theorems, computing the largest pole of zeta function,
\begin{center}
$\displaystyle  J(z)=\int_{U(\delta)} H(g(u),b,c)^{z} \varphi_{0} \vert g'(u) \lambda du'dbdc $.\\
\end{center}
In Watanabe is shown that this toric resolution is not complete and is necessary other resolution to the Kullback distance $ H(g(u),b,c) $ applying Hilbert basis again, now the monomial transformation is given by,
\begin{center}
$ g :\lbrace u_{kp},v_{k}; 1\leqslant k \leqslant K; 1 \leqslant p \leqslant M  \rbrace  \mapsto \lbrace b_{kp},c_{k}; 1 \leqslant k \leqslant K; 1 \leqslant p \leqslant M \rbrace$.
\end{center}
that is defined by,
\begin{center}
$ b_{11}=u_{11} $\\
$\qquad \qquad \qquad \qquad  b_{kp}=u_{11}u_{kp}, \quad (k\neq 1)$ o $ (p\neq 1) $,\\
$ c_{k}= u_{11}v_{k} $.
\end{center}
Then by Atiyah-Hironaka theorem exists analytic function $ H_{2}(a,u',v) $, such that,
\begin{center}
$ H(a,b,c)= u^{2}_{11}H(a,u',v) $,
\end{center}
which implies;
therefore $ \lambda_{1} \leqslant (M + 1)K/2 $.\\
Combining the results of above, the largest pole $ -\lambda_{1} $ of the poles of $ J(z)$ satisfies of inequality,
\begin{center}
$ \lambda_{1} \leqslant \dfrac{K}{2} min\lbrace N,M +1 \rbrace $,
\end{center}
With this information and corollary  \ref{Corollary 4.3.1}, we have the learning curve associated to the perceptron:\\
\begin{center}
$ K(n) \leqslant \dfrac{K}{2} min\lbrace N,M+1  \rbrace \log(n) + o\left( \log (n)\right)  $.
\end{center}
We reproduce the first toric resolution in the first re parametrization of the Kulback  distance and its toric variety associated by means of Hilbert basis; and the second resolution is given in Watanabe \cite{SWatanabe42001}; but is possible apply the technique with Hilbert basis the necessary times to up having the wished resolution, agreement to the Hironaka's theorem, \cite{HHironaka1964}.

\thispagestyle{plain}
\section*{\begin{center}
Conclusions.
\end{center}}
 The principal conclusion of this work is the use of Theorem \ref{Theorem 2.3.0.}; as consequence, Theorem \ref{Corollary 4.3.3}, that are fundamentals for the formalization and for reproducing of results previously reported in statistical singular learning, S. Watanabe \cite{SWatanabe12001}, \cite{SWatanabe42001}, \cite{SWatanabe52001}. The practical applications of these theorems is obtained by means of the use of Hilbert basis with Singular program, ref., \cite{DGPS}. This open the doors to look for other perspectives of investigation for machines with a high dimensional parameter space important in data science.It should be clear that the algorithmic complexity for computing Hilbert basis is a topic of current interest in computational algebraic geometry, but its solution for lattice polytope with thousands of vertex may well help in the solution of many other problems beyond the examples presented here.\\\\

\input amsart-template.bbl

\end{document}

\section{Introduction.}

The modelling of genetic-metabolic networks allows us to simulate and study the mechanisms of gene regulation, and its interplay with the environment. Gene regulation can be represented as an interaction graph in which each molecule is represented as a node and the interaction between them as links on the graph. The molecules depicted can be genes,  mRNA's, proteins or metabolites. Every interaction can represent a catalytic event, the repression or activation of a transcription factor, or the formation of a complex.

The lack of information about the kinetic parameters and the orders of the biochemical reactions involved in gene regulation and metabolic processes imposes constraints to solve analytically networks with a huge amount of nodes. Nevertheless, there are mathematical tools to approximate the parameter space. For instance, the parameter-space for an S-system can be delimited by linear optimization numerical techniques by identifying parameters intervals in order to approximate the parameter-space.

In the present study, we propose that a genetic-metabolic network, represented as a set of S-systems, can be solved using tools from algebraic geometry. First, we enunciate two theorems to formalize the S-Systems. In these theorems we show that the S-Systems are phenotypic toric varieties. We prove that a phenotypic toric variety is a generalization of a phenotypic polytope. Also, we define a lattice polytope, which is another combinatorial object associated with an S-system and which is required to do the geometrical realization of these algebraic objects. Also, we reproduce the solutions obtained for a set of S-Systems associated with a genetic-metabolic network in steady-state, previously studied by M. Savageau.

The first part of the work contains some background needed for the proof of our theorems. This includes a few concepts from Convex Geometry, such as Gordans Lemma, and also some Combinatorial Toric Algebraic Geometry. In particular, we state the definition of Toric Variety as an Algebraic Afine Scheme, since this definition is important in our formalization of S-Systems. Also, we introduce the main concepts of power law modelling of  biochemical systems (power law, dominant terms, phenotypic polytope), specifically in genetic-metabolic networks as studied by M. Savageau, \cite{MASavageau1976}. We prove that S-systems are toric varieties , both, assuming steady-state by demonstrating the Theorem 4.4-(embedding of S-Systems in toric varieties ), and beyond steady state by demonstrating Theorem 4.5-(toric resolution on S-Systems ).

In the second part, we study a gene network corresponding to a synthetic oscillator studied by M. Savageau \cite{LomnitzSavageau2014}. The network is composed of two genes whose expression is regulated by two transcription factors (TF's). The system is studied under different control modes, i.e; two negative feedback loops, two positive feedback loops, and a nested positive and negative feedback loops. The solution for this system shows that certain geometrical regions of the parameter-space are associated to each gene regulation control mode. Interestingly, most of the geometrical locus of the parameter-space is preserved, and this persistent region is associated with the oscillatory solution of the model. This persistent region is known as a phenotypic polytope, and the oscillatory solution is also called an oscillatory phenotype. Analytically, an oscillatory solution is a complex linear combination of sin and cosine functions varying over time, whose values represent the change of concentrations over time for every biochemical species in the network.

There are several important implications from  the formalization of the S-Systems as toric varieties. An embedding of any S-System into an algebraic torus must exist. A phenoptypic toric variety, which is an object more general than a phenotypic polytope, can be constructed from minimal objects called phenotypic toric ideals. The phenotypic toric ideals can be computed using Hilbert basis. The toric generators contain all the solution space associated with the S-system; because of this, the oscilattory phenotypes seen by Savageau are reproduced  and some new phenotypic polytopes are found. Finally, the toric generators and the use of Hironaka-Atiyah's theorem allow the study of the dynamics of any set of S-Systems.

\section{Combinatorial Toric Algebraic Geometry Background.}

In this section, we explain the background from combinatorial toric algebraic geometry, which we will need for the proof of our main theorems in section 3. Set $ M \, \subset \, \mathbb{R}^{n} $, by conv(\textbf{M}), we mean the convex hull of $ M $ , which is the set of all convex combinations of elements of $ M $. Given a finite subset M of $\mathbb{R}^{n}$, a \textbf{convex polytope} or \textbf{polytope} is the convex hull of M.
\begin{definition}\label{Definition 1.0} \cite{DCoxLittleSchenck2010}. A \textbf{lattice} $ N $ is a free abelian group of finite rank, and if its rank is $ n \, \in \, \mathbb{N} $, then $ N $ is isomorphic to $\mathbb{Z}^{n} $.\end{definition}

In particular for the case of each biochemical network, is determined by kinetic orders of reaction, which are determined by dissociation constants that are related to the concentrations of reactants to the substrates and products. We define a \textbf{condition lattice vector} $ p_{i}=(g_{i1},...,g_{im},h_{i1},...,h_{im}) \, \in\,\mathbb{Z}^{2m} $ for some integer $ i $; in the case that the $ g_{ij}$ and $ h_{ij} $ are rational numbers, we always can multiply them by a real number $ \lambda \, \in \, \mathbb{R} $, such that $ \lambda*p_{i}=\lambda*(g_{i1},...,g_{im},h_{i1},...,h_{im})  \, \in\,\mathbb{Z}^{2m} $, where each term is related to enviromental conditions related to gene regulation. \\
The set of all condition lattice vectors $ p_{i} \, \in\,\mathbb{Z}^{2m} $ form a finite convex hull set, hence a convex polytope. We are interested in the algebraic geometry properties of these objects.\\

\begin{definition}\label{Definition 1.1} \cite{GEwald1993}. A semi-group, is a non empty set \textbf{S} with an associative operation,
\begin{center}
$ + :  $\textbf{S}$ \times $\textbf{S}$ \longrightarrow $\textbf{S}.
\end{center}
A semi-group is named a \textbf{monoid}, its operation is commutative and if there exists a neutral element, i.e. an element $ 0 \, \in \,  $ \textbf{S}, where, $ s+0 =s,\, \forall \, s \, \in \, $ \textbf{S}, and this satisfies a cancellation law.\end{definition}
\begin{definition}\label{Definition 1.2.}
Given a cone $ \sigma=Con(p_{1},...,p_{n}) \, \subseteq \, \mathbb{R}^{n}$, if it is generated by lattice vectors, i.e., each $ p_{i} \,\in  \, \mathbb{Z}^{n}$ for all $ i $, we say that it is a \textbf{lattice cone}. And the convex hull of these lattice vectors, $ P= $conv$(p_{1},...,p_{n})$, we call a \textbf{lattice-polytope}.

\end{definition}

\textbf{Lemma 2.1. \cite{GEwald1993}.} Let $ \sigma\, \subset \, \mathbb{R}^{n} $ be a lattice cone, then $ \sigma \cap \mathbb{Z}^{n} $ is a monoid, see \cite{GEwald1993}.\\

\textbf{Lemma 2.2.} \textbf{(Gordan's Lemma)}. \cite{GEwald1993}. Let $ \sigma \, \subset \, \mathbb{R}^{n} $ be a lattice cone, then the monoid $ \sigma \cap \mathbb{Z}^{n} $ is finitely generated.\\
Now given the set of condition lattice vectors, $ p_{i}=(g_{i1},...,g_{im},h_{i1},...,h_{im}) \, \in\,\mathbb{Z}^{2m} $, with $ i=1,...,n $, we construct the lattice cone $ \sigma_{i} $. For some $ i $ and its associated monoid $S_{\sigma} = \sigma^{\vee} \cap \mathbb{Z}^{n}$ only intersecting  with $\mathbb{Z}^{n}  $ for lemma 2.1 (Gordan's lemma)
this monoid is therefore finitely generated, in the perspective of biochemical reactions there exists a set of minimal reactions associated to this minimal set (Hilbert basis) of kinetic orders, these minimal sets give us an interesting way to solve large-scale metabolic networks.\\

For topological reasons of completeness and compactness we will work not in the cone but in the dual cone associated to it. The definition of the dual cone is as follows.

\begin{definition}\label{Definition 1.3.} The dual lattice cone $\sigma^{\vee} \,\subseteq \, \mathbb{Z}^{m} $ associated to the cone $ \sigma \, \subseteq \, \mathbb{Z}^{n}$, is given by,
\begin{center}
$\sigma^{\vee} = \lbrace m \in \sigma: \langle m,u \rangle > 0, \, \forall u \, \in \sigma \rbrace $.
\end{center}
And also the equality of sets is true; $(\sigma^{\vee})^{\vee}= \sigma $.
\end{definition}
Also we can see of the definition, that if $ \sigma \,\subseteq \, \mathbb{Z}^{m} $ is a lattice cone, so $ \sigma^{\vee}  $ it is, and the same way the previous lemmas are applied on it. \\
Now we introduce one the most important concept in this work, as we see in the following.\\

The spectrum \textbf{Spec} of a ring $ R $, or algebraic affine scheme \textbf{Spec}$ (R) $ consists of all prime ideals of $ R $. In this work, we focus on the ring of polynomials of Laurent; i.e., $ \mathbb{C}[x,x^{-1}] $. And so, we construct the coordinate ring $ R_{\sigma} $
in the following manner. We take the support of any polynomial $ g \,\in \, \mathbb{C}[x,x^{-1}] $, and denote $ supp(g) $, which forms a lattice cone, called the Newton polyhedron or exponent-space. For some cone lattice $ \sigma \, \subset \, \mathbb{R}^{n} $, we have $R_{\sigma}= \left\lbrace g \, \in \, \mathbb{C}[x,x^{-1}] \, : \, supp(g) \, \subset \, \sigma^{\vee} \right\rbrace   $.

\begin{definition}\label{Definition2.} Let $\sigma$  be a lattice cone, the   \textbf{affine algebraic scheme:}
\begin{center}
$  X_{\sigma}=$ \textbf{Spec} $ (R_{\sigma}) $.
\end{center}
is called an \textbf{abstract toric affine variety} or \textbf{embedding of torus.}\\
\end{definition}

\section{Toric resolution and toric morphisms.}

Hibert basis in an important tool in Algebraic Geometry, \cite{BSturmfels1996}. In our case,
the Hilbert basis associated to a monoid $ N\, \subset \, \mathbb{Z}^{n} $, will enable us to compute explicitly a toric resolution in our main theorem. Also, its utilization will have important practical applications in the modelling of the dynamic solution and the steady-state of genetic-metabolic networks.

\begin{lemma}\label{Lemma 3.0.}\textbf{(Hilbert basis)}, \cite{BSturmfels1996}, \cite{DCoxLittleSchenck2010}, \cite{DCoxLittleOshea2004}.  Set $ \sigma \, \subseteq \, N $, then $ \sigma $ is an n-dimensional cone if and only if it is a strongly convex cone; i.e., $ \sigma \cap (-\sigma^{\vee}) ={0}$. In this case the monoid $ S_{\sigma} $ has a finite minimal set of generators $ H\, \subseteq \, M \, \simeq \, \mathbb{Z}^{d} $. \end{lemma}

In toric algebraic geometry it is a common practice to compute toric ideals from Hilbert basis. We will use the singular program for these computations. The following proposition and definition, will allow us to identify when a set of binomials are toric. These kind of binomials are important in our work, because the genetic metabolic networks can be represented in this way.

\textbf{Proposition 3.0.}\label{Proposition 2.0.} \cite{DCoxLittleSchenck2010}. Let  $ I $ be an ideal of the affine toric variety $ X_{\sigma} \, \subseteq \, \mathbb{C}^{n}$. Then this ideal has a representation given by:
\begin{center}
$  I(X_{\sigma})= \langle \, t^{l_{+}} - t^{l_{-}} \vert l \, \in L \, \rangle =\langle \, t^{\alpha} - t^{\beta} \vert \alpha - \beta \, \in L, \, \alpha, \, \beta \in \, \mathbb{Z}_{+}^{n} \, \rangle $,
\end{center}
where $ L $ is the kernel of the following morphism $ a_{i} \longrightarrow t^{a_{i}}$ with $ a_{i} \, \in \, \mathbb{Z}^{m}  $, $i=1,...,n  $.

\begin{definition}\label{Definition 2.1.3.} Let be $ L \, \subseteq \, \mathbb{Z}^{n} $, a sub-lattice.\\
\textbf{(a).} The ideal $ I_{L} = \langle \,  t^{\alpha} - t^{\beta} \vert \alpha - \beta \, \in L, \, \alpha, \, \beta \in \, \mathbb{Z}_{+}^{n}  \, \rangle$, is called a \textbf{lattice ideal}.\\
\textbf{(b).} A prime lattice ideal is called a \textbf{toric ideal.}\\\end{definition}
\textbf{Proposition 3.1.}\label{Proposition 2.1.} \cite{DCoxLittleSchenck2010}. An ideal $ I \, \subseteq \, \mathbb{C}[t_{1},...,t_{n}] $ is toric if and only if it is prime and it is generated by binomials.

\subsection{Toric Morphisms.}

We introduce in this section the concept of toric morphism. We will use this object in the conception of toric resolutions for singularities on a non-regular or singular variety.\\

\begin{definition}\label{Definition 2.3.0.} \cite{GEwald1993}. Let  $ \Phi :\mathbb{C}^{k} \longrightarrow \Phi(\mathbb{C}^{k}) $ be a monomial map, i.e., each non-zero component of $ \Phi $ is a monomial with coordinates in $ \mathbb{C}^{k} $, let $ X_{\sigma} \hookrightarrow  \mathbb{C}^{k}$ and $ X_{\sigma'} \hookrightarrow  \mathbb{C}^{m}$  be inclusions of toric affine varieties. If $ \Phi (X_{\sigma}) \, \subset \, X_{\sigma'} $, then $ \varphi  := \Phi \vert _{X_{\sigma}}$ is called a \textbf{toric affine morphism} of $ X_{\sigma} $  to $ X_{\sigma'} $. If $ \varphi $ is  bijective and its  inverse map $ \varphi^{-1} :X_{\sigma'} \longrightarrow X_{\sigma} $ is also a toric morphism, then  $ \varphi $ is called an \textbf{affine toric isomorphism} and it is denoted by $ X_{\sigma} \simeq X_{\sigma'} $.
\end{definition}

Recall  that a  \textbf{complex projective n-space $ \mathbb{CP}^{n} $} is the space of class of equivalence of pairs of points such that it consists of lines on $ \mathbb{CP}^{n}=\mathbb{C}^{n+1} / \sim  $. The relationship between points $ \sim $ is of the following manner. Given any vector $ v:= (\eta_{0},...,\eta_{n}) $ it defines a line $ \mathbb{C}*v $ and two of said vectors $ v \sim v' \in \mathbb{C}^{n+1}\setminus \lbrace 0 \rbrace $ define the same line if and only if, one is a multiple scalar of the other.\\

We say, that when there exists an inverse toric morphism in any point $ p \, \in \, X_{\sigma'} $, this toric morphism defines a \textbf{toric resolution} on $ p $ or toric blow up for distinguishing it from a blowing up usual, and these maps will coincide. Taking $ X_{\sigma'} \simeq \mathbb{C}^{2} \times \mathbb{CP}^{2} $, we will see an explicit toric resolution in the k-dimensional complex projective-space $ \mathbb{CP}^{k-1}  $ in the last theorem \ref{Theorem of toric modification} which uses the Hironaka-Atiyah's theorem. Both of these theorems, are important tools for toric resolutions of genetic-metabolic networks.

\begin{lemma}\label{Lemma 3.1.0}\cite{GEwald1993}. Let $ \sigma $  be a lattice cone and set $ \tau \, \preceq \, \sigma  $ a face. Then there exists a natural identification, of toric varieties, given by the toric morphism,
\begin{center}
$  X_{\tau^{\vee}} \: \simeq \: X_{\sigma^{\vee}} \setminus \lbrace  u_{k}= 0 \rbrace$.
\end{center}
where $ u_{k} $ is the last generator of the representation of the coordinate ring associated to $ X_{\sigma^{\vee}} $.\end{lemma}
\begin{definition} \label{Definition 2.3.3.1.} The isomorphism,
\begin{center}
$ \psi_{\sigma, \sigma'} : \,  X_{\sigma^{\vee}} \setminus \lbrace  u_{k}= 0 \rbrace \longrightarrow  X_{\sigma'^{\vee}} \setminus \lbrace  v_{l}= 0 \rbrace $.
\end{center}
is called \textbf{gluing morphism}, which glues the varieties $ X_{\sigma^{\vee}} $ and $ X_{\sigma'^{\vee}} $ in the variety $ X_{\tau^{\vee}} $.\end{definition}

With the following theorem, we aim to highlight the joint the importance of the concepts; lattice polytope, polynomial $ \mathbb{C}- $algebras and toric variety. And of the morphisms associated to them, for a novel representation and study of genetic-metabolic networks.

 ′
\begin{theorem}\label{Theorem 2.3.0.} \cite{MPCastilloOscar2017} and \cite{GEwald1993}. Set $ \sigma \, \subset \mathbb{R}^{n}= lin(\sigma) $ and $ \sigma' \, \subset \mathbb{R}^{m}= lin(\sigma') $, then  the following conditions are equivalent:
\begin{center}
\textbf{(a).} $ \sigma \, \simeq \, \sigma' $  {} {} {}  \textbf{(b).} $ R_{\sigma} \simeq R_{\sigma'} $  {} {} {} \textbf{(c).} $ X_{\sigma} \simeq X_{\sigma'} $.
\end{center}
\end{theorem}
The implications a) $ \Rightarrow $ b)$ \Rightarrow $ c) are proven by means of the following diagram and, previously, we prove that it is commutative.
\begin{center}

$ \qquad \qquad \sigma \quad \longrightarrow \quad R_{\sigma}  \quad \hookrightarrow\quad X_{\sigma}=$\textbf{Spec}$(R_{\sigma})$\\
$ \qquad \qquad \downarrow \uparrow  L^{-1}  \qquad  $ $ \downarrow \uparrow   \psi^{-1} \qquad \quad \quad\quad\quad $ $\downarrow \uparrow \varphi^{-1} $\\
$ \qquad \qquad \sigma^{'} \quad \longrightarrow \quad R_{\sigma^{'}} \quad \hookrightarrow\quad X_{\sigma^{'}}=$\textbf{Spec}$(R_{\sigma^{'}})$.
\end{center} ́
In the next section, we will point out where the principal concepts of systems biology are localized in Savageau, taking as a base the previous scheme.

In this work, we will combine the algebraic geometry concepts defined in the diagram above, with regulatory-metabolic networks defined as sets of S-systems. We represent an S-system as a set of non-linear algebraic equations. These equations represent the change of concentrations of some species in the genetic-metabolic network over time. In general, these equations can be written in the \textbf{formalism of power law}, where we invite you to seen in the beginning of section four.\\
Where $\dot{X}_{i}  $ denotes the change of concentration of i-esim specie. And in agreement with our scheme above $\dot{X}_{i}$ is an element of the coordinate ring, i.e; $\dot{X}_{i} \, \in \, R_{\sigma} $ the S-Systems in general live in this ring. But in this polynomial ring. In general, the S-Systems exist in this ring. Any polynomial ring, including the particular ring previously described, can be singular. We can see in the following demonstration how any S-system is singular., i.e.
\begin{center}
\[ \dfrac{\partial \dot{X}_{i}}{ \partial X_{j}}(0,...0)= 0,\]
\end{center}
without loss of generality for the zero concentration vector $  \bar{0}=(0,...,0) \, \in \, \mathbb{R}^{m}$, that represents the moment in which the concentration for any species is 0. In this lattice vector is clear to see the singularities $ \forall \, i=1,...,n $, it is always possible to prove by a translation of zero vector, for any point $ p \, \in \, \mathbb{R}^{m} $. This is an important fact for the dynamical solution in general. This problem combined with the lack of information about kinetic parameters and orders of reactions imply that for some values, some of the studied polynomials can tend to infinity. This is translated into a bad computational behaviour and a  non-obvious, or impossible biological situation. To solve this S-system, we translate this polynomial that belongs to the ring $ R_{\sigma} $.\\
So we have the polynomial $\dot{X}_{i} \, \in \, R_{\sigma} $, which is the singular case.
And now, to solve and characterize the S-Systems in a way most suitable. We translate this polynomial into a lattice-polytope utilizing the concept of Newton polyhedron, which consists of the lattice vectors of kinetic orders associated to an exponent-space. Formally we call it, the support of the polynomial, and we denote by $supp( \dot{X}_{i} )$, which in most of our example of S-Systems consist of two sums of binomials. For example, lets solve a genetic-metabolic network of five nodes. X is the concentration vector of all the nodes, so  $ X=(X_{1}, X_{2}, X_{3}, X_{4}, X_{5}) \, \in \, \mathbb{R}^{5} $. Now, let us consider only one equation of the system. \\
\begin{center}
   \[ \dot{X}_{3}=\alpha_{12} X_{1}^{g_{13}} X_{2}^{g_{12}} -  X_{3}^{h_{33}}.\]
\end{center}
Here $ \dot{X}_{3} \, \in \, R_{\sigma} $ since that the support associated to it, is
\begin{center}
$ supp( \dot{X}_{3} )=\lbrace (g_{13}, g_{12},0,0,0), \, (0, 0, h_{33}, 0, 0) \, \in \, \mathbb{Z}^{5} \,\, \rbrace $,
\end{center}
at the same time the lattice vectors generate the cone $ \sigma $ and we denote $ \sigma=Con((g_{13}, g_{12},0,0,0), (0, 0, h_{33}, 0, 0)) $. Now, we have translated the problem  into a lattice cone, such that $supp( \dot{X}_{3} ) \, \in \, \sigma  $, the geometrical properties of lattice cone are limited algebraically, but we provide this linear space with the characteristics of a monoid by intersecting it with the lattice group $ \mathbb{Z}^{5}  $, so $ M=\sigma \cap \mathbb{Z}^{5} $.\\
Next, we will explain the construction of the toric variety associated to an S-system, a very interesting object. We take the two lattice vectors of support, we have $ (g_{13}, g_{12},0,0,0)=g_{13}e_{1} + g_{12}e_{2} $ and $ (0, 0, h_{33}, 0, 0)=h_{33}e_{3} $, we associate the monomials to this lattice vector as: $  X^{g_{13}e_{1} + g_{12}e_{2}}=X_{1}^{g_{13}} X_{2}^{g_{12}}$ and $ X^{h_{33}} =X_{3}^{h_{33}}$. The toric variety associated to this terms are the primes monomials, which are denoted by $ X_{\sigma}= \lbrace X_{1}^{g_{13}}, X_{2}^{g_{12}}, X_{3}^{h_{33}} \rbrace$. This equivalent to say that this set is the algebraic affine scheme associated to the polynomial coordinate ring generated for the monomials, we write that as: $ X_{\sigma^{\vee}} =$\textbf{Spec}$  \mathbb{C}[X_{1}^{g_{13}}, X_{2}^{g_{12}}, X_{3}^{h_{33}}]= $\textbf{Spec}$ (R_{\sigma}) $.\\

The next figure summarizes the facts mentioned previously. It is a schematic representation of the morphism associated to the commutative diagram shown in Theorem 3.3. In the Figure 1.0 we can see the problems generated when there are singularities in any of three different levels. In the middle of the figure the set of equations of any S-system associated to a gene-metabolic network is represented. This set belongs to a finitely generated $ \mathbb{C} $-algebra which has some singular polynomials. One can take the prime polynomials associated to these singular polynomials to construct an irregular toric variety or not complete variety. Also, we can construct the irregular lattice polytope (left side of the figure) from the support space of these singular polynomials, also called exponent-space or Newton polyhedron. The lattice vectors associated as vertex of these irregular lattice polytope, are represented with a matrix array in row, this representation induces a linear transformation $ L $, in the dictionary of linear transformations. When the lattice polytope be regular then the matrix associated $ L_{A} $ to this, will have the property of uni-modularity, i.e., \textbf{det}$ L_{A}=\pm 1 $, and the linear transformation $ L $ now is named as uni-modular transformation. It is work of us to transform  the irregular lattice polytope to a regular lattice polytope by means of uni-modular transformation and these are done by means of Hilbert basis generated from the set of vertex of irregular polytope.

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=2.5in]{cd.png}}
\caption{\label{fig_simvarex} Three different representations of singular S-Systems and three representations for non-singular S-systems.}
\end{figure}
The problem of having singularities is that they increase the complexity of the system, making it more difficult to solve systems with a large number of equations, such as large-scale genetic-metabolic networks. This problem can be tackled using tools from algebraic geometry. This can be solved by mapping any of these objects: irregular lattice-polytope, singular S-systems polynomials, or irregular toric varieties, into regular or non-singular forms.\\
 This solution is represented in the next figure, in which each irregular object is mapped to its regular or non-singular counterpart.

The principal methodology developed in our work transforms irregular lattice polytopes into regular or non-singular polytopes. This will allows us to transform a set of singular polynomials, that represent S-systems into a set of non singular polynomials of S-Systems. This will open the possibility to model large-scale metabolic networks.\\
In our work, we study the irregular lattice-polytope (Figure 1, top left panel). We transform the irregular lattice cone into a monoid by intersecting with $ \mathbb{Z}^{n} $, which can be written as $ M= \sigma\cap \mathbb{Z}^{n} $. By applying the Gordan's lemma to the monoid we can conclude that this is finitely generated, that is to say there exist a set of generators such that they generate all M. This set of generators is minimal and is named the Hilbert basis. The elements of Hilbert basis are lattice vectors and form a regular or non-singular
lattice polytope (Figure 1, bottom left panel). By arranging these vectors into a matrix, we induced a uni-modular transformation, $ L $. i.e; \textbf{det}$ L_{A}=\pm 1 $. This transformation and its inverse transformation $ L^{-1} $  are used to map irregular lattice-polytopes into regular lattice-polytopes (Figure 1, arrows between top and bottom panel).\\
This mapping is the  algorithmic basis to computationally transform irregular lattice-polytopes into regular ones.

\begin{theorem} \label{theorem 4.} (\textbf{\textit{Hironaka-Atiyah}}), \cite{MFAtiyah1970}. Let $ g $ be a real analytical function $ (\neq 0) $ in a neighbourhood of $ p_{0} \, \in \, \mathbb{R}^{n} $. Then there exist an open set $\omega \, \in \, W \subset \, \mathbb{R}^{n} $, a real analytical variety $ W_{0} $ and a proper analytical map $ \varphi :W_{0}$ to $W$ such that:\\
i). $ \varphi :(W_{0}-A_{0}) \longrightarrow (W-A) $ is an isomorphism, where  $ A =g^{-1}(p_{0}) $ and $ A_{0}=\varphi^{-1}(A)=(g(\varphi))^{-1}(p_{0}) $;\\
ii). For each $ \omega  \, \in W_{0} $, there are local analytical coordinates  $ (\omega_{1},....,\omega_{n}) $ such that there exists a resolution map, we write $ g(\varphi(\omega))=\pm h_{1,...,n} \omega_{1}^{r_{1}}*\omega_{2}^{r_{2}}*...*\omega_{n}^{r_{n}} $, where $ r_{1},....,r_{n} $ are  non negative integers and $ h_{1,...,n} $ is an invertible analytic function; see ref.
\end{theorem}

The theorem is a modified version of the theorem "resolution of singularities" and is used for analytical varieties. The original theorem was proved by Hironaka, \cite{HHironaka1964}.\\

In the following we enunciate a general result for  non regular or singular varieties.
\begin{theorem}\label{Theorem of toric modification} \cite{GEwald1993}. Let $ X_{\sum} $ be a regular toric variety, and let $ X_{\sum_{0}} $ be a toric invariant sub variety defined by the star
$ st(\sigma,\sum) \simeq  \sum_{0}  $ of
$ \sigma $ into $ \sum $;
$ 1 < k:=dim \sigma \leq n $.\\
Then under toric blow up $ \psi^{-1}_{\sigma} $, any point $ x \in X_{\sum_{0}} $ is  substituted by a k-dimensional \textbf{(k-1) projective space.}\\
See ref. for the proof of this fact.
\end{theorem}
This theorem says that if we have a singular variety in any point $ p \, \in \,\mathbb{R}^{m} $, i.e. the condition is true. So, the variety $ X_{\sigma_{0}} $  built from its support polynomial $ \dot{X}_{i}=0 $, is a non regular toric variety, which implies that it has singularities. By Hironaka's theorem  \cite{HHironaka1964}, there exists a morphism called resolution or toric blow-up, which yields a resolution of these singularities. Using this morphism the variety $ X_{\sigma_{0}} $ is mapped  into a regular toric variety $ X_{\sigma}$ or non singular variety. It is possible to show that the transformation of coordinates consists in mapping each singularity into a complex n-dimensional (n-1)-projective space as pointed in the last theorem \label{Theorem of toric modification}, i.e. we take the coordinates of singular point and we substitute them by a dimensional (k-1)-projective space, where $ k $ depends on the dimension, see \cite{GEwald1993}. The parameter space is of special interest for the study of the dynamics of gene regulation, as we will see.

\begin{comment}
We begin our work, considering a convex set $K \subseteq \mathbb{R}^{n} $ with $ K \, \neq \, \emptyset $ and we form all convex combinations of elements into it, is to say, its hull convex. If $ K $ is finite, we say that this hull convex is a polytope.

\begin{definition}\label{Definition1.}

A \textbf{lattice polytope} or \textbf{lattice cone} $ \sigma $ is a hull convex, which is a free abelian group of finite rank, if it is of rank $ n $, then is isomorphic to $ \mathbb{Z}^{n} $; given the case for $ p \, \in \, \mathbb{Z}^{n} $, we also write $ \sigma=Con(p) $, to see other definitions; \cite{DCoxLittleSchenck2010} and \cite{GEwald1993}.
\end{definition}

And now, we affirm the following facts and for the details of its proofs, we recommended see the references, only we do a compilation of these results for the demonstrations in the last theorems.\\
.
\textbf{ Affirmation 1}. Let $ \sigma \, \subset \, \mathbb{R}^{n} $ be a lattice cone, then $ \sigma \cap \mathbb{Z}^{n} $ is a \textbf{monoid}, \cite{GEwald1993}.\\

\textbf{Affirmation 2}. Let $ \sigma \, \subset \, \mathbb{R}^{n} $ be a lattice cone,   then the monoid $ \sigma \cap \mathbb{Z}^{n} $ is  finitely generated, \textbf{\textit{Gordan's Lemma}}, \cite{GEwald1993}.\\

\textbf{Affirmation 3.}. Let $ \sigma \subset \, \mathbb{R}^{n} $ be a  $ n $ dimensional  lattice cone with $0 \preceq \sigma $ where $ \preceq $ denote the equivalence relationship of \textbf{face}, and let $ n_{1},...,n_{r} $ be  the inner normal facets of $ \sigma $. Then
\begin{center}
$  \sigma^{\vee}= Con(n_{1},...,n_{r})$.
\end{center}
also we say that $\sigma^{\vee}$ is the \textbf{dual cone} of $ \sigma $, and is clear to prove the sets equality $(\sigma^{\vee})^{\vee}=\sigma  $, for the demonstrations of the affirmation, see \cite{GEwald1993}.\\

Now we remember that the spectrum \textbf{Spec} of a ring $ R $, or algebraic affine scheme \textbf{Spec}$ (R) $ consists of all prime ideals of $ R $. For our aim, we work particularly in the ring of polynomials of Laurent; i.e., $ \mathbb{C}[x,x^{-1}] $. And so, we construct the coordinate ring $ R_{\sigma} $
in the following manner; we take the support of any polynomial $ g \,\in \, \mathbb{C}[x,x^{-1}] $, and we denote $ supp(g) $, which form a lattice cone, called the Newton polyhedron or exponent-space, for some cone lattice $ \sigma \, \subset \, \mathbb{R}^{n} $, we have $R_{\sigma}= \left\lbrace g \, \in \, \mathbb{C}[x,x^{-1}] \, : \, supp(g) \, \subset \, \sigma \right\rbrace   $.

\begin{definition}\label{Definition2.} Let $\sigma$  be a lattice cone, the   \textbf{affine algebraic scheme:}
\begin{center}
$  X_{\sigma}=$ \textbf{Spec} $ (R_{\sigma}) $.
\end{center}
is called a \textbf{abstract toric affine variety} or \textbf{embedding of torus.}\\
\end{definition}

\textbf{Proposition 2.0.}\label{Proposition 2.0.} Let $ \sigma \subseteq \, M\, \subset \, \mathbb{Z}^{n}$ be a lattice cone and $ \sigma^{\vee} \, \subset \, N$ the dual lattice cone associated, with $M,N$ finite sets of lattice vectors, then the monoid,

\begin{center}
$ S_{\sigma}=\sigma^{\vee} \cap N$.
\end{center}
is finitely generated and these lattice vectors are minimals, and we denote this set $ H $,\begin{proof} see in \cite{BSturmfels1996}, \cite{DCoxLittleSchenck2010}, details of the demonstration.\end{proof}
The set $ H $ are the \textbf{basis Hilbert}, and computational algorithms are the great importance for us, are applied persistently on toric S-Systems.

\begin{theorem} Let $ I $ be an ideal of the toric variety $ X_{\sigma} \, \subseteq \, \mathbb{C}^{r}$ and we consider the monoids homomorphism,, $ S \, \subset \, \mathbb{N}^{n}$, so:
\begin{center}
$ \phi \, : S \, \longrightarrow \, \mathbb{Z}^{r}$ with $ \phi(s)=\sum_{i=1}^{n} \lambda_{i} s_{i}$.
\end{center}
Then we define,
\begin{center}
$  I(X_{\sigma})= \langle \, x^{s_{+}} - x^{s_{-}} \vert s \, \in S \, \rangle =\langle \, x^{a} - x^{b} \vert a - b \, \in S, \, a, \, b \in \, \mathbb{Z}_{+}^{n} \, \rangle $,
\end{center}
where $ S $ is the kernel of the homomorphism $\phi $.\end{theorem}
\begin{proof}

 See for the proof of the proposition, ref. \cite{DCoxLittleSchenck2010}.\end{proof}

If the ideal $ I_{S} = \langle \,  x^{a} - x^{b} \vert a - b \, \in S, \, a, \, b \in \, \mathbb{Z}_{+}^{n}  \, \rangle$, we call it a \textbf{lattice ideal}, and if is prime, we say that it is a \textbf{toric ideal}.\\
\begin{theorem}
Let $ I \, \subseteq \, \mathbb{C}[x_{1},...,x_{n}] $ be toric ideal if and only if it is generated by binomials, and it is prime. \end{theorem}
\begin{proof} details of the proof, see ref. \cite{DCoxLittleSchenck2010}.\end{proof}

\begin{theorem}\label{theorem 3.} Let $ \sigma \, \subset \mathbb{R}^{n}= lin(\sigma) $ be and $ \sigma_{0} \, \subset \mathbb{R}^{m}= lin(\sigma_{0}) $, then  the following conditions are equivalent, here $ lin $ means the minimal space that contains the lattice cones respectively:
\begin{center}
\textbf{(a).} $ \sigma \, \simeq \, \sigma_{0} $  {} {} {}  \textbf{(b).} $ R_{\sigma} \simeq R_{\sigma_{0}} $  {} {} {} \textbf{(c).} $ X_{\sigma} \simeq X_{\sigma_{0}} $.
\end{center}
\end{theorem}
\begin{proof}

for details of this proof, see refs. \cite{MPCastilloOscar2017} and \cite{GEwald1993}. \end{proof}

\begin{theorem} \label{theorem 4.} (\textbf{\textit{Hironaka-Atiyah}}) Let $ g $ be a real analytical function $ (\neq 0) $ in a neighbourhood of $ p_{0} \, \in \, \mathbb{R}^{n} $. Then there exist an open set $\omega \, \in \, W \subset \, \mathbb{R}^{n} $, a real analytical variety $ W_{0} $ and a proper analytical map $ \varphi :W_{0}$ to $W$ such that:\\
i). $ \varphi :(W_{0}-A_{0}) \longrightarrow (W-A) $ is an isomorphism, where  $ A =g^{-1}(p_{0}) $ and $ A_{0}=\varphi^{-1}(A)=(g(\varphi))^{-1}(p_{0}) $;\\
ii). For each $ \omega  \, \in W_{0} $, there are local analytical coordinates  $ (\omega_{1},....,\omega_{n}) $ such that exist a resolution map, we write $ g(\varphi(\omega))=\pm h_{1,...,n} \omega_{1}^{r_{1}}*\omega_{2}^{r_{2}}*...*\omega_{n}^{r_{n}} $, where $ r_{1},....,r_{n} $ are  non negative integers and $ h_{1,...,n} $ is an invertible analytic function; see ref. \cite{MFAtiyah1970}.
\end{theorem}
And we conclude this section commenting that given a $ X_{\sigma_{0}} $ non regular toric variety, is to say with singularities, by Hironaka's theorem  \cite{HHironaka1964}, there exist a morphism called resolution or toric blow-up for our case, and consists of resolution of these singularities; the variety $ X_{\sigma_{0}} $ is mapped  in a regular toric variety $ X_{\sigma}$ or non singular variety, is possible showing, see \cite{GEwald1993}, that the transformation of coordinates consists in to map each singularity into to a complex n-dimensional (n-1)-projective space, these spaces are of our special interest for the study of dynamical analysis in gene regulation, as we can see.
\end{comment}
\section{Biochemical Machines: Toric resolution in genetic-metabolic networks.}
In this section we introduce biochemical systems particularly genetic-metabolic networks modelled using the power law formalism of enzyme kinetics, defined and widely studied by M. Savageau, \cite{MASavageau1976}, \cite{MASavageauVoit1987}, \cite{MASavageau2001}. Subsequently, we prove that they are toric varieties.\\\\

\textbf{Formalism of power law for biochemical systems.}\\
The power law formalism is a particular representation of enzyme kinetics, formally a Taylor's expansion in several variables capable of describing biochemical reactions of metabolic and gene regulatory mechanisms, formulated as rational functions produced by the balance of mass equations.
\begin{center}
\[ \dot{X}_{i}= \sum_{k=1}^{p_{i}} \alpha_{ik} \prod_{j=1}^{m}X_{j}^{g_{ijk}} - \sum_{k=1}^{q_{i}} \beta_{ik}\prod_{j=1}^{m} X_{j}^{h_{ijk}},\: i=1,...,n.\]
\end{center}
This expression represents the basic principle of mass conservation, called \textbf{general mass action (GMA)}, in the form of the power law formalism. Each term is associated to elements of a network of reactions, where  $\dot{X}_{i} $ represents a derivative of $ X_{i} $ over time t. This expression represents the rate law of degradation or production of any $ i $-esim species $ X_{i} $ (gene, mRNA, proteins or metabolites) in the network. This variation in time describe the flux of biomass throughout all elements in any biochemical reaction and genetic network. The first sum is that of the products that contribute to the production to the species $ X_{i} $ and the terms of the second sum are elements of consumption due to the degradation or dilution of $ X_{i} $, \cite{MASavageau1976}, \cite{MASavageau2001}.\\
\begin{definition}\label{Definition 6.0.}(\textit{\textbf{Dominant terms}}). A dominant term is defined
as the largest term of a given sign for an equation of the GMA-system (equation defined above); the dominant terms with positive and negative signs are the \textbf{dominant positive term} and the \textbf{dominant negative term}, respectively, see ref. \cite{MASavageau1976}.\end{definition} \

\begin{definition}\label{Definition 6.1.} (\textit{\textbf{S-systems}}). Using the concept defined above, we can have a finite number of combinations of dominant terms, exactly  \[ \prod_{j=1}^{m} p_{j}q_{j} ; \]  partitions of the space of concentrations $(X_{1},...,X_{n}) $, where each partition is a dominant sub-system and it is associated with a particular system of equations, called \textbf{S-systems}, M. Savageau, \cite{MASavageauVoit1987}, \cite{MASavageau1976}, \cite{LomnitzSavageau2013}, \cite{LomnitzSavageau2014}, \cite{LomnitzSavageau2015}; as shown below
\[ \dot{X}_{i}=  \alpha_{ip_{i}} \prod_{j=1}^{m} X_{j}^{g_{ip_{i}}} -  \beta_{iq_{i}} \prod_{j=1}^{m}X_{j}^{h_{iq_{i}}}, \, i=1,...,n. \].\\
for details, see M. Savageau \cite{MASavageauVoit1987}.\end{definition}\
Note in this last definition that the sums were neglected, at difference to the formalism power law formalism complete. An S-System only consist of dominant terms.
\begin{definition}\label{Definition 6.2.} (\textit{\textbf{Phenotypic Polytope}}). A phenotype is defined as a set of concentrations or fluxes corresponding to a valid combination of dominant terms. They define boundary conditions obtained from the solution of S-systems under steady-state, in logarithmic-space. Each boundary condition defines an m-dimensional half-space and the intersection of these half-spaces yields an m-dimensional \textbf{phenotypic polytope}. \end{definition}\
The phenotypic polytope associated with the S-Systems can be considered to be a realization of a lattice polytope, taking the support of each S-systems related to the kinetic orders of the metabolic reactions.\\
Using the previous definitions, in the following Theorem we prove that the S-Systems are toric varieties.
\begin{theorem}\label{Theorem 1.} (\textit{\textbf{Toric Embedding of S-systems}})
 Let $ \dot{X}_{i} $ be an S-system, with initial conditions $ X_{i}(0)=X_{io} \,\in\, \mathbb{R}$, for $ i=1,...,n $.\\

Then the S-system $ \dot{X}_{i}$, for all $ i=1,2,...,n $, is generated by \textbf{toric ideals} $ I_{H}$ and as a consequence, each $ \sigma_{i} \subseteq \, \mathbb{Z}^{n} $ \textbf{phenotypic polytope} is associated to an affine toric variety, namely,
\begin{center}
$ X_{\sigma^{\vee}_{i}} =$\textbf{Spec} $( R_{\sigma_{i}})$, $ \: \forall \, i=1,2,...,n,  j=1,...,m $.\\

\end{center}
With coordinate ring $ R_{\sigma_{i}} $ associated to the monomial $ \mathbb{C}- $algebra, $ \mathbb{C}[X_{j}^{g_{ij}}, X_{j}^{h_{ij}}] $, and it is called a \textbf{toric phenotypic variety} or simply \textbf{phenotypic variety}.
\end{theorem}
\begin{proof}
 We prove the result by construction, given the following polynomial  $ \dot{X}_{i} =\alpha_{i} \prod_{j=1}^{m}X_{j}^{g_{ij}} -  \beta_{i}\prod_{j=1}^{m}X_{j}^{h_{ij}} $, for some $ i $ which is an S-system, where $\dot{X}_{i} \, \in \, \mathbb{C}[X_{j}^{g_{ij}}, X_{j}^{h_{ij}}] $. It is clear to see that the monomial $\mathbb{C} -$algebra is generated by the monomials $ X_{j}^{g_{ij}} $ and $ X_{j}^{h_{ij}} $; and  always an S-system is defined by two positive condition lattice vectors $ a_{i}=(g_{i1},...,g_{im}), \: b_{i}=(h_{i1},...,h_{im}) \: \in \: \mathbb{Z}_{+}^{m} $. Allow these vectors of kinetic orders which are associated to each binomial of each equation of an S-system consist of positive terms, otherwise,  $ g_{ij} $ and $ h_{ij} $ could be rational numbers or they could be negative integers. In steady-state, i.e., $ \dot{X}_{i}=0 $, one can always multiply by units of the ring $ X^{\pm r} $ and $ X^{\pm s} $ the polynomial  $ \dot{X}_{i} =0$, with $ r, s \: \in \, \mathbb{Z} $, such that, $ rsa_{i}, \: rsb_{i}\: \in \: \mathbb{Z}_{+}^{m} $ or equivalently once that we prove the S-Systems are toric, we see that exists an invariance for action torus ($ T \times X_{\sigma^{\vee}} \longrightarrow X_{\sigma^{\vee}} $, given by $ (t,x)\longmapsto t*x=(t^{a_{1}}*x_{1},...,t^{a_{k}}*x_{k})$ with $t^{a_{i}}\, \in\, \mathbb{C}^{*}  $).\\

Now set $ \sigma $ to be the lattice cone generated by the two rays $ a_{i} $ and $ b_{i} $. We will prove that its dual lattice cone is strongly convex, i.e., it complies with the property $ \sigma^{\vee} \cap (-\sigma^{\vee})=\lbrace 0 \rbrace $, so that we can apply the Hilbert basis's lemma. First, we note that the lattice vector $ a_{i}- b_{i} \,\in\, \sigma$ if and only if $ a_{i} \preceq b_{i}  $ where $ \preceq $ is a partial order induced by the inclusion on the faces of the lattice cone, and therefore the zero lattice vector is a face, $ 0\preceq \sigma $. This assumption is true, if and only if $\sigma^{\vee}  $ is full dimensional, i.e., that it contains bases as vector space to generate an octant of n-dimensional Euclidian-space; and if and only if $\sigma^{\vee}  $ is strongly convex. Note that, all these facts are standard results of convex geometry.\\
Since $ \sigma^{\vee}_{i} \, \subseteq \, \mathbb{Z}^{m} $ is strongly convex, we can use the Hilbert basis theorem to obtain a finite minimal set of generators for the monoid $ S_{\sigma_{i}}= \sigma_{i}^{\vee} \cap \mathbb{Z}^{m} $. We order them in a row matrix array  $ H=\lbrace h_{ij} \rbrace $ (Hilbert basis). The ideals associated to them are:
\begin{center}
$ I_{H}=< X^{a'_{i}}- X^{b'_{i}} \, \vert \, (a'_{i} - b'_{i}) \, \in \, ker(H) > $,
\end{center}
These ideals are primes and generate $ \dot{X}_{i} $, to show that are primes, we chose any binomial, and is sufficient to show that it is prime, suppose that it is not, so it is factorized at least for two elements in the $ \mathbb{C}-$algebra, i.e., $x^{a}-x^{b}=f*g  $, we construct the Newton polyhedron, taking $supp(x^{a}-x^{b})=supp(f*g)=supp(f)+supp(g)$ (property of Minkowski sum) \cite{BSturmfels1996}, therefore the elements that generate the lattice cones $supp(f)$ and $ supp(g) $, also generate the lattice cone $supp(x^{a}-x^{b})=Con(a,b)$, but $ a,b \,\in\,H$ are minimal elements, from here the contradiction, therefore $x^{a}-x^{b}$ is prime. Now the application of the definition \ref{Definition 2.1.3.} implies that they are lattice ideals, in agreement with the hypothesis of the proposition 3.0. We conclude that they also are toric ideals. Now, we construct the regular lattice cone $ \sigma^{\vee}_{iH} $ associated to these Hilbert basis, where the regularity is provided  by the fact that, det$ (H)=\pm 1 $, this process is valid for each $ i=1,...,n $. With these facts and using the theorem \ref{Theorem 2.3.0.}, we construct the coordinate ring $ R_{\sigma^{\vee}_{Hi}}= \lbrace \dot{X}_{Hi} \, \in \,   \mathbb{C}[\bar{X_{j}}^{g_{ij}},\bar{X_{j}}^{h_{ij}}] \vert \: supp(\dot{X}_{Hi}) \subset  \sigma^{\vee}_{Hi} \rbrace $, with lattice cone $ \sigma^{\vee}_{Hi}= Con (H) $ generated by the Hilbert basis, where we made a change of coordinates of the binomials $ \dot{X}_{i} $ in the system provided by the Hilbert basis, and we denote them now by $\dot{X}_{iH}$. This change of coordinates is always possible using the morphisms $ L_{H} $ and $ \chi $ defined in theorem \ref{Theorem 2.3.0.}. Also there exists a monomial homomorphism $ j_{\sigma^{\vee}_{Hi}} $ that give rise for all $ i$, into the affine algebraic scheme $ X_{\sigma^{\vee}_{Hi}}= $ \textbf{Spec} ($ R_{\sigma^{\vee}_{Hi}} $). This is a toric variety with monomial $\mathbb{C} -$algebra given by the isomorphisms classes $ \mathbb{C}[S_{\sigma^{\vee}_{Hi}}]=\mathbb{C}[X_{j}^{g_{ij}}, X_{j}^{h_{ij}}] / I_{H} \simeq  \mathbb{C}[\bar{X_{j}}^{g_{ij}},\bar{X_{j}}^{h_{ij}}]$ (toric re-parametrization), with new coordinate system given by the classes prime monomials $\bar{X_{j}}^{g_{ij}}  $ and $\bar{X_{j}}^{h_{ij}}  $. In this way, we have proven that  \textbf{S-systems are toric} or have an embedding in the algebraic torus, and $ X_{\sigma^{\vee}_{Hi}} $ is the associated \textbf{phenotypic variety}; with this fact we conclude the proof. \end{proof}

S-systems are usually solved under steady-state conditions. In the following, we make a toric resolution in S-Systems without assuming steady-state. Theorem \ref{Theorem 2.} will enable the study of their dynamics.

\begin{theorem}\label{Theorem 2.}\textbf{(\textit{Toric resolution in S-systems})}. Let $ \dot{X}_{i} $ be an S-system.
Then there are maps of resolution $ \psi_{i} : X_{\sigma_{i}^{\vee}} \longrightarrow X_{\sigma_{i}^{' \vee}}$ for each $ i $ (toric morphisms) such that, they are a re-parametrization in the coordinate ring $ \mathbb{C}[\omega_{1},...,\omega_{m}] $ thus;
\begin{center}
$\dot{X}_{i}(\psi_{i}(\omega) )=h_{i} \omega_{1}^{\alpha_{1}}*.....*\omega^{\alpha_{m}}_{m}  $, with $ \omega_{i}(0)=\omega_{i0} $.
\end{center}
where the $h_{\alpha_{1},...,\alpha_{m}} $ are units of the ring, with $ \omega=(\omega_{1},...,\omega_{m}) \, \in  \, \mathbb{R}^{m}$ and  $ \alpha_{1} > 0,...., \alpha_{m} > 0 $, positive integers.\\
\end{theorem}
\begin{proof}
The proof is given by construction and by applying  Theorems 3.6, and the theorem of Hironaka-Atiyah, as we will see. First we take for some $ i $, with $ i=1,...,n $ an equation of an given S-system $\dot{X}_{i}  $, in agreement with  Theorem 3.6, let $\dot{X}_{i} \, \in \, R_{\sigma}  $.The definition of $R_{\sigma}  $ implies that:
\begin{center}
$supp( \dot{X}_{i})=\lbrace a_{i} \rbrace=\lbrace (g_{i1},...,g_{im}),(h_{i1},...,h_{im}) \rbrace \, \subseteq \, \sigma$;
\end{center}
we form the lattice cone $ \sigma=Con\left( (g_{i1},...,g_{im})-(h_{i1},...,h_{im})\right)  \,\subseteq \, \mathbb{Z}^{m} $ that consists, of the difference lattice vector, which implies the partial order $ \preceq $. i.e. Given $g_{i}=g_{i1},...,g_{im} $ and $ h_{i}=h_{i1},...,h_{im} $, set $ g \preceq h $ or vice versa, if and only if $ (g-h) \,\in\, \sigma $ for some lattice cone $ \sigma $, this construction of cones is important,  because agreement of definition 3.1 and Proposition 2.0, the difference $ g -h$ can be lifting to toric binomials $ (X^{g}- X^{h}) $, and we show it by the following process.\\

Given the lattice cone $ \sigma \,\subseteq \, \mathbb{Z}^{m} $ built as above. From the same Theorem 3.6, it follows the existence of the following morphisms,
$ \chi_{i} \, : \sigma \, \longrightarrow  \, R_{\sigma}$, $ \chi^{-1}_{i} \, :R_{\sigma} \, \longrightarrow  \, \sigma $, $ L_{H} \, :\sigma \, \longrightarrow  \, \sigma'$, $ \chi_{i}' \, : \sigma' \, \longrightarrow  \, R_{\sigma'}$, $ \psi_{i} \, :R_{\sigma}  \, \longrightarrow  \, R_{\sigma'}$, $ \psi^{-1}_{i} \, :  R_{\sigma'} \, \longrightarrow  \,  R_{\sigma}$, where the $\psi_{i}  $ are morphisms of $ \mathbb{C} $-algebras.\\

Explicitly we have the maps, $ \chi_{i}(a_{i})=X^{a_{i}} \, \in \, R_{\sigma} $; $ \chi^{-1}_{i}(X^{a_{i}})=a_{i} \, \in \, \sigma $; $ L_{H}(a_{i})=h_{i} \, \in \, \sigma' $; $ \chi_{i}'(h_{i})=\omega^{h_{i}}=\omega_{1}^{h_{i1}}*...*\omega_{n}^{h_{im}} \, \in \, R_{\sigma'} $; $ \psi^{-1}_{i}(\omega)=\omega^{h_{i}}\, \in \, R_{\sigma} $, where $ a_{i}=(a_{i1},...,a_{im}) \, \in \, \mathbb{Z}^{m} $ is the condition lattice vector associated to $\dot{X}_{i}$, we also write $X^{a_{i}}=X_{1}^{a_{i1}}*...*X_{m}^{a_{im}} $. The map $ L_{H} $ represent the linear transformation of the matrix $ H=(h_{ij})$, this matrix is an array row, this consist of the Hilbert basis computed on the monoid associated to the non-regular lattice cone $ \sigma $, i.e, $ S_{\sigma}=\sigma^{\vee} \cap  \mathbb{Z}^{m}$. \\
Once we have computed the Hilbert basis on the monoid $ S_{\sigma} $, the matrix array induces an uni-modular transformation, i.e., det$H=\pm 1 $ (if $ m=n $, if $ m \neq n $ then we take squares sub-matrix), which is the condition for regular lattice cones, these basis $ H $ conform a new lattice cone $ \sigma' \,\subseteq \, \mathbb{Z}^{m} $. In this form we obtain the map of lattice cones $ L(\sigma)=\sigma' $, where $ \sigma' $ now represents a regular cone. From the commutativity property illustrated in the diagram of Theorem 3.6, we have the composition of maps,
\begin{center}
$ \psi^{-1}_{i} \, \circ \, \chi' = \chi \, \circ \, L_{H}^{-1}$.
\end{center}
from the equality of maps, we can obtain the following:
\begin{center}
$ \psi^{-1}_{i}(\chi'(\omega))=\omega^{h_{i}}=\omega_{1}^{h_{i1}}*...*\omega_{n}^{h_{im}}=\chi( L_{H}^{-1}(h_{i}))=X^{L_{H}^{-1}(h_{i})} $.
\end{center}
Therefore it is a new coordinate system induced from Hilbert basis, and we can re-parametrize each monomial of $ \dot{X}_{i} $, hence:
\begin{center}
$ \dot{X}_{i}(\psi^{-1}_{i}(\omega))=\alpha_{i}\omega_{1}^{h_{i1}}*...*\omega_{n}^{h_{im}}-
\beta_{i}\omega_{1}^{h'_{i1}}*...*\omega_{n}^{h'_{im}}   $.
\end{center}
some monomials can be factorized, because some integers can be $h_{ik} \geq h'_{il} $ or vice verse, for some $ k,l \,\in \, \lbrace 1,...m \rbrace$, i.e; if there exist terms with $ s_{ij}+p_{ij}=h_{ij} $ and $s_{ij}+q_{ij} = h'_{ij}$, then,
\begin{center}
$ \dot{X}_{i}(\psi^{-1}_{i}(\omega))=\omega_{1}^{s_{i1}}*...*\omega_{n}^{s_{ir}}(
\alpha_{i}\omega_{1}^{p_{i1}}*...*\omega_{n}^{p_{il}}-
\beta_{i}\omega_{1}^{q_{i1}}*...*\omega_{n}^{q_{im}} )  $.
\end{center}
The re-parametrization process is well defined. During this process we will apply Hironaka-Atiyah's theorem, given that the polynomial $ \dot{X}_{i} $ is an analytical function; if the polynomial $ \dot{X}_{i}(\psi^{-1}_{i}(\omega)) $ is a non singular polynomial the toric resolution
is completed; and we redefine $ f_{1,...,m}= \omega_{1}^{s_{i1}}*...*\omega_{n}^{s_{ir}}$ as an analytical invertible function, and  positive integers $ \alpha_{1}=s_{i1} >0,...,\alpha_{m}=s_{i1}>0 $, such that,
\begin{center}
$ \dot{X}_{i}(\psi^{-1}_{i}(\omega))= f_{1,...,m}\omega_{1}^{\alpha_{1}}*...*\omega_{n}^{\alpha_{m}}$.
\end{center}
Otherwise, if the resulting polynomial is singular, we repeat the process: We compute the Hilbert basis on the lattice cone of the support of the new polynomial $  \dot{X}_{i}(\psi^{-1}_{i}(\omega))$, until a non-singular polynomial is generated. The process is finite, and the maps are composed of finite number of times. The finiteness of the process is guaranteed by the Hironaka's theorem in its form of Principal Theorem I, which is a widely known theorem and it is not shown in this work, see \cite{HHironaka1964}. We also have that the morphism $(\psi^{-1}_{i}(\omega))  $ induce a toric morphism and yields a non singular toric variety, namely $ X_{\sigma'}= $\textbf{Spec}$(R_{\sigma})  $, whenever $\psi^{-1}  $ is non singular too. And with these facts, we conclude the proof. \end{proof}

\section{Applications of Toric S-Systems.}

\textbf{Phenotypes in an oscillatory synthetic gene network.}\\

 In this example we will describe a gene network of two genes with two regulators. This  network is subject to different control modes of transcription, i.e., the genes are expressed by different mechanisms, repression, activation and a combination of them. The example is represented by the following algebraic system of non linear differential equations. In Figure 2, a schematic for the studied gene network is shown. $ X_{1} $ is an activator mRNA, $ X_{2} $ is the activator protein, $ X_{3} $ is the repressor mRNA, $ X_{4} $ is the repressor protein, $ X_{A} $ is the inmature activator and $ X_{R} $ is the immature repressor., see M. Savageau, \cite{LomnitzSavageau2015}.\\
\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=2.0in]{co1.png}}
\caption{\label{fig_simvarex} Schematic representation of oscillatory gene circuit with six elements.}
\end{figure}

In Savageau and Lomnitz \cite{LomnitzSavageau2015}, the following variables are considered as follows: $X_{2i}=x_{2i}/(K^{0}_{2i}), \: i=1,2.$, and\\
\begin{center}
$ X_{2i-1}= \left(\dfrac{(\beta^{0}_{2i-1})(\sqrt{\rho^{0}_{2i-1}})}{\alpha^{0}_{2i-1}} \right)x_{2i-1}, \, i=1,2.$,

\end{center}

where here the values of the concentrations of the immature activator and repressor were normalized with respect to its respectively values in steady-state, i.e, when we clear terms to solve equations in steady state, we have:\\

\begin{center}

$ X_{A}=\left[ \dfrac{(\beta^{0}_{1} + \mu^{0})(\beta^{0}_{A} +\mu^{0}) \sqrt{\rho^{0}_{1}}}{(\alpha^{0}_{1})(\alpha^{0}_{A})} \right] x_{A} $,

$ X_{R}=\left[\dfrac{(\beta^{0}_{3} + \mu^{0})(\beta^{0}_{R} + \mu^{0}) \sqrt{\rho^{0}_{3}}}{(\alpha^{0}_{3})(\alpha^{0}_{R})} \right] x_{R} $,
\end{center}
the regulators are normalized with respect to their dissociation constants of DNA. Near, their promoter regions, the same pair of binding sites for the repressor-DNA, and activator-DNA at both promoters are assumed, i.e, $ K_{2R}=K^{0}_{2} $ and $K_{4A}=K^{0}_{4} $, the parameters are explained below.\\

Now we give the explicit system of differential algebraic equations that represent the S-System associated to our gene-metabolic network , where this non linear system captures the advantages of essential non linear behaviour locally, the principal details about how to obtain this S-System can be consulted in Savageau and Lomintz \cite{LomnitzSavageau2015}. We here, only focus in analysing it by means of toric geometry.\\
This S-systems is a valid approximation of the full system if the conditions following
for the dominant terms are satisfied:\\\\

$ \dot{X}_{1}= \left[ \alpha_{1} (\beta^{1}_{0} + \mu^{0}) \sqrt{\rho^{0}_{1}} (\alpha_{1}^{0})^{-1} \right] \delta_{1}(K_{2})^{-g_{12}}(K^{0}_{2})^{g_{12}}X_{2}^{g_{12}} (K_{4A})^{g_{14}} (K_{4}^{0})^{-g_{14}} X_{4}^{-g_{14} } - (\beta_{1} + \mu) X_{1} $,\\\\
$ \dot{X}_{A}= (\beta^{0}_{A} + \mu^{0})\left[ \dfrac{\alpha_{A}}{\alpha^{0}_{A}} \right]  X_{1} - (\beta_{A} + \mu) X_{A}$,\\\\
$\dot{X}_{2} = (\beta^{0}_{2} + \mu^{0}) \left( \dfrac{\beta_{A}}{\beta^{0}_{A}}\dfrac{\alpha_{1}^{0} n_{2}^{0}}{(\beta_{1}^{0} + \mu^{0})\sqrt{\rho_{1}^{0}} K_{2}^{0}} \right) X_{A} - (\beta_{2} + \mu)X_{2} $,\\\\\\
$ \dot{X}_{3}= \left[  \alpha_{3} (\beta^{3}_{0} + \mu^{0}) \sqrt{\rho^{0}_{3}} (\alpha_{3}^{0})^{-1} \right]
(K_{2R})^{-g_{32}}(K_{2}^{0})^{g_{32}} (X_{2})^{g_{32}} - (\beta_{3}+ \mu) X_{3} $,\\\\
$ \dot{X}_{R}= (\beta^{0}_{R} + \mu^{0})\left[ \dfrac{\alpha_{R}}{\alpha^{0}_{R}} \right]  X_{1} - (\beta_{R} + \mu) X_{R}$,\\\\
$ \dot{X}_{4} = (\beta^{0}_{4} + \mu^{0}) \left( \dfrac{\beta_{R}}{\beta^{0}_{R}}\dfrac{\alpha_{3}^{0} n_{4}^{0}}{(\beta_{3}^{0} + \mu^{0})\sqrt{\rho_{3}^{0}} K_{4}^{0}} \right) X_{R} - (\beta_{4} + \mu)X_{4} $.\\\\\\

where $ K_{2R} $ and $ K_{4A}$ are defined as:\\\\\\

$ K_{4A}= \dfrac{\left[  \dfrac{\alpha_{3}^{0} n_{4}^{0}}{(\beta_{3}^{0} + \mu^{0})\sqrt{\rho_{3}^{0}}} \right]}{\left\lbrace   \left( \delta_{1} \left[ \dfrac{\alpha_{1}^{0} n_{2}^{0}}{(\beta_{1}^{0} + \mu^{0})\sqrt{\rho_{1}^{0}} K_{2}^{0}} \right]^{g_{12}}  + 1 - \pi_{1}  \right) \sqrt{\rho_{1}^{0}} - \pi_{1} \right\rbrace^{\dfrac{1}{g_{14}}}} $;\\\\

$ K_{2R}=\dfrac{   \left[  \dfrac{ \alpha_{1}^{0} n_{2}^{0}  }{(\beta_{1}^{0}  + \mu^{0}) \sqrt{\rho_{1}^{0}} } \right] \sqrt{\rho_{3}^{0}}^{\dfrac{1}{g_{32}}}}{ \left\lbrace   \delta_{3} \left[  \dfrac{\alpha_{3}^{0} n_{4}^{0}}{(\beta_{3}^{0}  + \mu^{0})\sqrt{\rho_{3}^{0}} K_{4}^{0} }  \right]^{g_{34}}   + (\pi_{3} - 1)\sqrt{\rho_{3}^{0}} + \pi_{3} \right\rbrace^{\dfrac{1}{g_{32}}}   } $. \\\\\\
The details of this formulation are given with a wide development of the calculations, in; J. Lomintz and M. Savageau, \cite{LomnitzSavageau2015}.\\

The parameters space and all variables belong to the following Euclidian-space, $ X=(X_{1}, X_{2}, X_{3}, X_{4}, X_{A}, X_{R}) \, \in \, \mathbb{R}^{6}  $; X represents the concentration vector, and it contains the concentration for each of the elements of the system.\\

\begin{itemize}
\item $ X_{1}: $ Concentration of activator mRNA.
\item $ X_{2}: $ Concentration of activator protein.
\item $ X_{3} :$ Concentration of repressor mRNA.
\item $ X_{4} :$ Concentration of repressor protein.
\item $ X_{A} :$ Concentration of form immature activator.
\item $ X_{R} :$ Concentration of form immature repressor.
\end{itemize}
For this system, the condition lattice vector is defined by $ g=(g_{12}, g_{14}, g_{32}, g_{34}) \, \in \, \mathbb{Z}^{4} $, where the $ g_{ij}'s $, are kinetic parameters that represent the cooperativity of repression or activation and are associated to the number of binding sites occupied by the regulator protein.\\
The constants $ \alpha_{i}, \beta_{i}, \mu^{0}, \rho_{i}^{0}, n_{i}^{0} \, \in \, \mathbb{R}$ are: maximum rates of transcription, rate constants of mRNA and tagged proteins at midlevel expression, rates of dilution due to exponential growth, regulatory capacities, and ratio of proteins to mRNA, respectively. The constants $K_{2}, K_{4}, K_{2R}, K_{4A}  \, \in \, \mathbb{R}$, are concentrations of regulator for half-maximal binding to control sequences in the DNA, $ K_{2} $ and $ K_{4} $ for binding near their own promoter regions and $ K_{2R} $, $ K_{4A} $ for binding near other promotors, see, J. Lomintz and M. Savageau \cite{LomnitzSavageau2014}.\\
The parameters $ \pi_{i}, \, \delta_{i}= 1,0. \, \in \, \mathbb{Z}_{2}$, for $ i=1,2,3,4. $ represent the different modes of control for the activator and repressor proteins, For $ \pi_{i}= 1 $, the gene network is considered to behave as an activator-primary system, in which the default mode of the system is inactive, and the activator is the primary regulator needed to increase the expression from the promoter. On the other hand, in the case $ \pi_{i}= 0$, the behavior of the system is as a repressror-primary system, in which the promoter is fully active in the absence of both regulators and the repression is needed to reduce expression from the promoter. $  \delta_{i}=1$ will imply a double-regulator network and , and $ \delta_{i}=0 $ will imply a single-regulator network. The solution for the S-system, described previously, under different control modes is shown in the following figures.

\begin{center}
\textbf{Gene circuit with a single negative feedback loop.}\\
$ \delta_{1}=0, \delta_{3}=0, \pi_{1}=0, \pi_{3}=1 $.

\begin{figure}[htbp]
\centering
\includegraphics[height=1.60in]{C1.png}
\caption{\label{fig_simvarex} Single negative feedback loop.}
\end{figure}
\end{center}

\begin{center}
\textbf{Gene circuits with one positive and one negative feedback loop.}\\

$ \delta_{1}=1, \delta_{3}=0, \pi_{1}=1, \pi_{3}=1 $,

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C2.png}}

\caption{\label{fig_simvarex} Activator-only control of repression transcription and activator-primary control of activator transcription.}

\end{figure}

\end{center}

\begin{center}
$ \delta_{1}=1, \delta_{3}=0, \pi_{1}=0, \pi_{3}=1 $.

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C3.png}}
\caption{\label{fig_simvarex} Activator-only control of repression transcription and repressor-primary control of activator transcription.}
\end{figure}

\end{center}

\begin{center}
\textbf{Gene circuits with one positive and one negative feedback loops.}\\

$ \delta_{1}=1, \delta_{3}=1, \pi_{1}=1, \pi_{3}=1 $,

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C4.png}}
\caption{\label{fig_simvarex} Activator-primary control of repressor transcription and activator-primary control of activator transcription.}
\end{figure}

\end{center}

\begin{center}
$ \delta_{1}=1, \delta_{3}=1, \pi_{1}=0, \pi_{3}=1 $.

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C5.png}}
\caption{\label{fig_simvarex} Activator-primary control of repressor transcription and repressor-primary control of activator transcription.}
\end{figure}

\end{center}

\begin{center}
\textbf{Gene circuits with one positive and nested negative feedback loops.}\\

$ \delta_{1}=1, \delta_{3}=1, \pi_{1}=1, \pi_{3}=0 $,

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C6.png}}
\caption{\label{fig_simvarex} Repressor-primary control of repressor transcription and activator-primary control of activator transcription.}
\end{figure}

\end{center}

\begin{center}
$ \delta_{1}=1, \delta_{3}=1, \pi_{1}=0, \pi_{3}=0 $.

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=1.60in]{C7.png}}
\caption{\label{fig_simvarex} Repressor-primary control of repressor transcription and repressor-primary control of activator transcription.}
\end{figure}

\end{center}

Each one of the different control modes (represented as a combination of $ \delta's $ and $ \pi's $) is mapped to complex geometric regions, which are coloured in yellow in Figures 3-9. In a study done by Lomnitz and Savageau \cite{LomnitzSavageau2014}, these regions correspond to one complex solution associated to an eigenvalue with real positive part. In Savageau's work, a specific region of the possible solution-space of this system corresponds to an oscillatory region. In our analysis, performed using Hilbert basis and toric ideals, we partially reproduce these results and we found that part of the oscillatory region is conserved across most of the different control modes, this conserved region is called a conserved \textbf{phenotypic polytope}. This technique is important since it would allow the modelling of gene circuits with hundreds of interacting nodes, with the possibility to find \textbf{conserved phenotype polytopes} within such regulatory networks.\\

\textbf{Methodological analysis of S-systems by means of toric varieties.}

We will start by considering an S-System in steady-state. In such state there is no change in the concentration for every species of the network, i.e. $  \dot{X}_{2} =\dot{X}_{4}=0 $. We are only interested in$ X_{2} $ and $ X_{4} $ since $ X_{2} $ is the activator protein and $ X_{4} $ is the repressor protein.
This set of polynomials is an $ \mathbb{C}-$\textbf{algebra} well defined, $ \mathbb{C}[X_{i}^{g_{ij}}, X_{i}^{h_{ij}}] $; by means of theorem 3.6\label{Theorem 2.3.0.}, the toric ideals are finitely generated and have also an associated geometric realization of lattice polytope as we will see. Our coordinate ring is defined in the variables $ X=(X_{1}, X_{2}, X_{3}, X_{4}, X_{A}, X_{R}) \, \in \, \mathbb{R}^{6}$ and is oriented with respect to a canonical basis $ B=\lbrace e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6} \rbrace \, \subseteq \, \mathbb{R}^{6}$. To construct the Newton polytope from the S-sytems, the next methodological development is used: \\
1). \textbf{Computing of the support-space for each equation of the S-System}, we have agreement of the definition of Newton-polyhedron or exponent-space associated to each equations to S-sytems, the following sets can be therefore stipulated:\\
$ supp(\dot{X}_{1}) =\left\lbrace g_{12}e_{2}-g_{14}e_{4}, e_{1} \right\rbrace $.\\
$ supp(\dot{X}_{A}) =\left\lbrace e_{1}, e_{5} \right\rbrace $.\\
$ supp(\dot{X}_{2}) =\left\lbrace e_{5}, e_{2} \right\rbrace $.\\
$ supp(\dot{X}_{3}) =\left\lbrace g_{32}e_{2}, e_{3}\right\rbrace $.\\
$ supp(\dot{X}_{R}) =\left\lbrace e_{3}, e_{6}\right\rbrace $.\\
$ supp(\dot{X}_{4}) =\left\lbrace e_{6}, e_{4}\right\rbrace $.\\\\

2.) \textbf{Computing of the Hilbert basis for the monoids associated to the dual lattice cones.} Remember that the basis were obtained with the Singular program. Is important to note that the Hilbert basis are invariant for all multiplicative constant, for more details of this argument, ref. \cite{DGPS}, for example, for the lattice vector ray $e_{2} \, \in \, \mathbb{R}^{6} $ belongs to the elongated lattice vector ray $ g_{12}e_{2} \, \in \, \mathbb{R}^{6}$ for some constant $ g_{12} \, \in \, \mathbb{R} $, as is the case of the parameter values $ g_{12}, g_{14}, g_{32} $ and without more problem we adopt the values $g_{12}=g_{14}=g_{32} = 1 $, and we have.\\

Monoid $S_{\sigma^{\vee}_{1}}= \sigma^{\vee}_{1} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{1}} =\left\lbrace  e_{1}, e_{2}- e_{4}\right\rbrace $.\\

Monoid $S_{\sigma^{\vee}_{A}}= \sigma^{\vee}_{1} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{A}} =\left\lbrace  e_{1}, e_{5}  \right\rbrace $.\\

Monoid $S_{\sigma^{\vee}_{2}}= \sigma^{\vee}_{2} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{2}} =\left\lbrace e_{5}, e_{2} \right\rbrace $.\\

Monoid $S_{\sigma^{\vee}_{3}}= \sigma^{\vee}_{3} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{3}} =\left\lbrace  e_{2}, e_{3} \right\rbrace $.\\

Monoid $S_{\sigma^{\vee}_{R}}= \sigma^{\vee}_{R} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{R}} =\left\lbrace  e_{3}, e_{6}  \right\rbrace $.\\

Monoid $S_{\sigma^{\vee}_{4}}= \sigma^{\vee}_{4} \cap \mathbb{Z}^{6} $, Hilbert basis associated, $ H_{\sigma^{\vee}_{4}} =\left\lbrace e_{6}, e_{4}  \right\rbrace $.\\

3). \textbf{Geometrical realizations of Newton polytopes and dual cones}. We write the notation of dualCone but we don't explicitly compute the lattice vectors that generate the dual cones, these are computed implicitly in the Singular program \cite{DGPS}. \\

$\sigma^{\vee}_{1}=$ dualCone $(g_{12}e_{2}-g_{14}e_{4}, e_{1}) $,\\
$ \sigma^{\vee}_{A}=$ dualCone $(e_{1}, e_{5} ) $,\\
$ \sigma^{\vee}_{2}=$ dualCone $(e_{5}, e_{2})$,\\
$ \sigma^{\vee}_{3}=$ dualCone $( g_{32}e_{2}, e_{3})$,\\
$ \sigma^{\vee}_{R}=$ dualCone $( e_{3}, e_{6}) $,\\
$ \sigma^{\vee}_{4}=$ dualCone $( e_{6}, e_{4})$.\\\\

Let us remember that the geometric realization of a toric variety is made by dual cones since in this space the cones have the same dimension, and the gluing maps are well defined, see, ref. \cite{GEwald1993}. The computing of the dual cones is easy to see from the Hilbert basis for few elements or from these toric ideals. Therefore, the dual cones are:\\\\
$\sigma^{\vee}_{1}=$ Con  $(e_{3}, e_{5}, e_{6}, e_{1}+e_{4}, e_{1}+e_{2}) $,\\
$\sigma^{\vee}_{A}=$ Con $(e_{2}, e_{3}, e_{4}, {e}_{6}, e_{1}+e_{5})$,\\
$\sigma^{\vee}_{2}=$ Con $(e_{1}, e_{3}, e_{4}, e_{6},e_{2}+e_{5})$,\\
$\sigma^{\vee}_{3}=$ Con $(e_{1}, e_{4}, e_{5}, e_{6}, e_{2}+e_{3} )$,\\
$\sigma^{\vee}_{R}=$ Con $(e_{1}, e_{2}, e_{4}, e_{5}, e_{2}+e_{6} ) $,\\
$\sigma^{\vee}_{4}=$ Con $(e_{1}, e_{2}, e_{3}, e_{5}, e_{3}+e_{6} )$.\\
These were calculated from the toric ideals, once computed in 5).\\

4.) \textbf{Computing of toric ideals associated to the Hilbert basis.}
The toric ideals for $ \langle \dot{X}_{A} \rangle,\: \langle \dot{X}_{R} \rangle,\: \langle \dot{X}_{1} \rangle,\: \langle \dot{X}_{2} \rangle, \: \langle \dot{X}_{3} \rangle,\: \langle \dot{X}_{4} \rangle$,  In these equations it is easy to calculate these values for some parameters $ h_{i,j}=1 $ for all $ i, j=1, 2, 3, 4, A, R $. Explicitly, the parameters can take the  values, for the same rationale applied in 2), $ g_{ij}= \lbrace g_{12}= g_{14}=g_{A1}=1, g_{2A}=1, g_{32}=g_{34}= g_{R3}=1, g_{4R}=1 \rbrace $ generating all the possible connections for different control modes in this gene circuit. And were computed by means of the technique of \textbf{Hilbert basis} in agreement with the theorems previously established (Theorem 4.4\label{Theorem 1.} and Theorem 4.5\label{Theorem 2.} (toric resolution in S-systems)). Their computing was made with the program, \textbf{Singular} \cite{DGPS}. Before it is necessary to renormalise the variables of S-system as input for the Singular program, hence:\\
We rename the parameters for simplicity and after rename the variables abusing of notation, being that the toric ideals associated to each equation are the same if the variable is multiplied for different parameter.\\
$\dot{X}_{1} =a*X_{2}X_{4}^{-1}- b*X_{1}  $,\\
$\dot{X}_{A} =c*X_{1}- d*X_{A} $,\\
$\dot{X}_{2} =e*X_{A}- f*X_{2} $,\\
$\dot{X}_{3} =g*X_{2}- h*X_{3} $,\\
$\dot{X}_{R} =i*X_{3}- j*X_{R} $,\\
$\dot{X}_{4} =k*X_{R}- l*X_{4} $.\\

And rename variables, also let us to do abuse of the notation here, for each variable $ u_{i} $, as we see.\\
$\dot{u}_{1} =u_{2}u_{4}^{-1}- u_{1}  $,\\
$\dot{u}_{A} =u_{1}- u_{A} $,\\
$\dot{u}_{2} =u_{A}- u_{2} $,\\
$\dot{u}_{3} =u_{2}- u_{3} $,\\
$\dot{u}_{R} =u_{3}- u_{R} $,\\
$\dot{u}_{4} =u_{R}- u_{4} $.\\
We do simply $\dot{u}_{i}= \dot{X}_{i}  $, being that finally will do $ \dot{X}_{i}=0 $ (steady-state condition), so the toric ideals are:\\
$I_{\sigma^{\vee}_{1}}=\langle \dot{X}_{1} \rangle= \langle u_{1}u_{4}-1, u_{1}u_{2}-1, u_{3}- 1, u_{A}-1, u_{R}-1 \rangle $,\\
$I_{\sigma^{\vee}_{A}}=\langle \dot{X}_{A} \rangle= \langle u_{1}u_{A}-1, u_{R}-1, u_{2}-1, u_{3}- 1, u_{4}- 1\rangle  $,\\
$I_{\sigma^{\vee}_{2}}=\langle \dot{X}_{2} \rangle= \langle u_{2}u_{A}-1, u_{1}-1, u_{R}-1, u_{3}- 1, u_{4}- 1 \rangle  $,\\
$I_{\sigma^{\vee}_{3}}=\langle \dot{X}_{3} \rangle= \langle u_{2}u_{3}-1, u_{1}-1, u_{R}- 1, u_{A}- 1, u_{4}- 1 \rangle  $,\\
$I_{\sigma^{\vee}_{R}}=\langle \dot{X}_{R} \rangle= \langle u_{2}u_{R}-1, u_{1}-1, u_{2}-1, u_{A}- 1, u_{4}- 1 \rangle  $,\\
$I_{\sigma^{\vee}_{4}}=\langle \dot{X}_{4} \rangle= \langle u_{4}u_{R}-1, u_{1}-1, u_{2}-1, u_{3}- 1, u_{A}- 1 \rangle  $.\\

Therefore the fan $ \sum=\lbrace \sigma^{\vee}_{i} : \, i=1,2,3,4,A,R \rbrace $ built with this system of lattice cones, yields the phenotypic variety given by:
\begin{center}
$  X_{\sigma^{\vee}_{i}} =$\textbf{Spec} $( \mathbb{C}[X_{j}^{g_{ij}}, X_{j}^{h_{ij}}])$, $ \: \forall \, i, j=1,2,3,4, A, R. $
\end{center}

5.) \textbf{Realization of the toric varieties associated to the dual cones.} The spectrum or set of prime ideals associated to the toric varieties, is our new coordinate system for translating and solving the S-system associated to any gene-metabolic network. In particular, we are interested in the gene circuit described earlier for reproducing the oscillatory phenotype associated to them, and given in Savageau \cite{LomnitzSavageau2014}.\\

More explicitly the toric varieties are:\\\
$X_{\sigma^{\vee}_{1}} =$\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{1} \cap \mathbb{Z}^{6}])=
$\textbf{Spec} $( \mathbb{C}[u_{1},u_{2}u^{-1}_{4}]) \\
=
\left\lbrace u_{1},u_{2}u^{-1}_{4} \right\rbrace \simeq \mathbb{C}_{(u_{1})}\times \mathbb{C}_{(u_{2})} \times \mathbb{CP}_{(u_{4})} $,\\
$  X_{\sigma^{\vee}_{2}} =$\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{2} \cap \mathbb{Z}^{6}])= $\textbf{Spec} $( \mathbb{C}[u_{2},u_{A}])\\
=\left\lbrace u_{2},u_{A} \right\rbrace \simeq \mathbb{C}^{2}_{(u_{2},u_{A})}$,\\
$  X_{\sigma^{\vee}_{3}} = $\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{3} \cap \mathbb{Z}^{6}])=$\textbf{Spec} $( \mathbb{C}[u_{2},u_{3}])\\
= \left\lbrace u_{2},u_{3}\right\rbrace \simeq \mathbb{C}^{2}_{(u_{2},u_{3})}$,\\
$  X_{\sigma^{\vee}_{4}} =$\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{4} \cap \mathbb{Z}^{6}])=$\textbf{Spec} $( \mathbb{C}[u_{4},u_{R}])\\
=\left\lbrace u_{4}, u_{R} \right\rbrace \simeq \mathbb{C}^{2}_{(u_{4},u_{R})}$,\\
$  X_{\sigma^{\vee}_{A}} =$\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{A} \cap \mathbb{Z}^{6}])=$\textbf{Spec} $( \mathbb{C}[u_{1}, u_{A}])\\
=\left\lbrace u_{1}, u_{A} \right\rbrace\simeq \mathbb{C}^{2}_{(u_{1},u_{A})}$,\\
$  X_{\sigma^{\vee}_{R}} =$\textbf{Spec} $( \mathbb{C}[\sigma^{\vee}_{R} \cap \mathbb{Z}^{6}])=$\textbf{Spec} $( \mathbb{C}[u_{3},u_{R}])\\
=\left\lbrace u_{3}, u_{R} \right\rbrace \simeq \mathbb{C}^{2}_{(u_{3},u_{R})}$.\\\\
\textbf{Gluing toric varieties.} Now we glue these toric varieties through of the gluing maps defined by means of their common faces in each one of the dual cones\\
\begin{center}
$ \psi_{\sigma_{1},\sigma_{A}}= X_{\sigma^{\vee}_{1}} / \lbrace u_{3}=u_{R}= 0\rbrace \longrightarrow X_{\tau_{1}}\longrightarrow   X_{\sigma^{\vee}_{A}} / \lbrace u_{3}=u_{R}= 0\rbrace $,\\

$ \psi_{\sigma_{A},\sigma_{2}}= X_{\sigma^{\vee}_{A}} / \lbrace u_{3}=u_{4}=u_{R}= 0\rbrace \longrightarrow X_{\tau_{A}}\longrightarrow   X_{\sigma^{\vee}_{2}} / \lbrace u_{3}=u_{4}=u_{R}= 0\rbrace $,\\

$ \psi_{\sigma_{2},\sigma_{3}}= X_{\sigma^{\vee}_{2}} / \lbrace u_{1}=u_{4}=u_{6}= 0\rbrace \longrightarrow X_{\tau_{2}}\longrightarrow   X_{\sigma^{\vee}_{3}} / \lbrace u_{1}=u_{4}=u_{6}= 0\rbrace $,\\

$ \psi_{\sigma_{3},\sigma_{R}}= X_{\sigma^{\vee}_{3}} / \lbrace u_{1}=u_{4}=u_{5}= 0\rbrace \longrightarrow X_{\tau_{3}}\longrightarrow   X_{\sigma^{\vee}_{R}} / \lbrace u_{1}=u_{4}=u_{5}= 0\rbrace $,\\
$ \psi_{\sigma_{R},\sigma_{4}}= X_{\sigma^{\vee}_{R}} / \lbrace u_{1}=u_{2}=u_{5}=0\rbrace \longrightarrow X_{\tau_{4}}\longrightarrow   X_{\sigma^{\vee}_{4}} / \lbrace u_{1}=u_{2}=u_{5}=0 \rbrace $,\\
$ \psi_{\sigma_{4},\sigma_{A}}= X_{\sigma^{\vee}_{4}} / \lbrace u_{3}=u_{5}= 0\rbrace \longrightarrow X_{\tau_{R}}\longrightarrow   X_{\sigma^{\vee}_{A}} / \lbrace u_{3}=u_{5}= 0\rbrace $.\\
\end{center}
Where the sub-varieties $ X_{\tau^{\vee}_{1}}$, $ X_{\tau^{\vee}_{A}}$, $ X_{\tau^{\vee}_{2}}$, $ X_{\tau^{\vee}_{3}}$, $ X_{\tau^{\vee}_{R}} $, $ X_{\tau^{\vee}_{4}}$ are given by:\\
$ X_{\tau_{1}}=X_{\sigma^{\vee}_{1}}/\lbrace z^{e_{1}}=u_{1}=0\rbrace \simeq \mathbb{C}^{2}_{(u_{3},u_{R})}\times \mathbb{C}_{(u_{1})}$,\\
$ X_{\tau_{A}}=X_{\sigma^{\vee}_{A}}/\lbrace z^{e_{5}}=u_{5}=0 \rbrace \simeq \mathbb{C}^{3}_{(u_{3},u_{4},u_{R})}\times \mathbb{C}_{(u_{5})}$,\\
$ X_{\tau_{2}}=X_{\sigma^{\vee}_{2}}/ \lbrace z^{e_{2}}=u_{2}=0 \rbrace \simeq \mathbb{C}^{3}_{(u_{1},u_{4},u_{6})}\times \mathbb{C}_{(u_{2})}$,\\
$X_{\tau_{3}}= X_{\sigma^{\vee}_{3}}/ \lbrace z^{e_{3}}=u_{3}=0 \rbrace \simeq \mathbb{C}^{3}_{(u_{1},u_{4},u_{5})}\times \mathbb{C}_{(u_{3})}$,\\
$ X_{\tau_{R}}=X_{\sigma^{\vee}_{R}}/ \lbrace z^{e_{6}}=u_{6}=0 \rbrace \simeq \mathbb{C}^{3}_{(u_{1},u_{2},u_{5})}\times \mathbb{C}_{(u_{6})}$,\\
$ X_{\tau_{4}}=X_{\sigma^{\vee}_{4}}/ \lbrace z^{-e_{4}}=u^{-1}_{4}=0 \rbrace \simeq \mathbb{C}^{2}_{(u_{3},u_{5})}\times \mathbb{C}^{*}_{(u^{-1}_{4})}$.

With common faces are given by the intersections in the generators of each monoid $S_{\sigma^{\vee}_{i}}=\sigma^{\vee}_{1}\cap \mathbb{Z}^{6}  $ with $ i=1,2,3,4,A;R $, i.e., common elements in the Hilbert basis, hence, $ \tau_{1}=\lbrace e_{1}\rbrace,\tau_{A}=\lbrace e_{5} \rbrace ,\tau_{2}=\lbrace e_{2} \rbrace ,\tau_{3}=\lbrace e_{3} \rbrace  ,\tau_{R}=\lbrace e_{6} \rbrace ,\tau_{4}=\lbrace e_{4}\rbrace $.\\

\begin{figure}[htbp]
\centering
\makebox{\includegraphics[height=2.2in]{gluing.png}}
\caption{\label{fig_simvarex} The dual cones $ \sigma^{\vee}_{1} $ and $ \sigma^{\vee}_{A} $ are gluing through the face generated by $ e_{3},e_{6} $, similar for the cones $ \sigma^{\vee}_{3} $ and $ \sigma^{\vee}_{R} $, the cone $ \sigma^{\vee}_{R} $ intersect for three hyperplanes to the dual cone $ \sigma^{\vee}_{3} $ and the edges $ e_{1},e_{4},e_{5} $ are common faces to them.}
\end{figure}

The \textbf{phenotypic variety} is obtained with these gluing maps, and the fan defined as $ \sum= \cup_{i} \sigma^{\vee}_{i}$, $ i=1,2,3,4,A,R. $. Then we have the following toric variety.
\begin{center}
$  X_{\sum} =X_{\sigma^{\vee}_{1}}\cup X_{\sigma^{\vee}_{2}}\cup X_{\sigma^{\vee}_{3}} \cup X_{\sigma^{\vee}_{4}} \cup X_{\sigma^{\vee}_{A}} \cup X_{\sigma^{\vee}_{R}}$.
\end{center}
Also is true the realization of general toric variety associated to the S-system in this case:
$ X_{\sum}= $\textbf{Spec} $( \mathbb{C}[\sum \cap \mathbb{Z}^{6}])=$\textbf{Spec}$( \mathbb{C}[u_{1},u_{2},u_{3},u_{4},u_{A},u_{R}]/ I_{\sum})\\
=\left\lbrace u_{2}=u^{2}_{4} \right\rbrace$.

Substituting all the algebraic relations, we obtain the parabolic relationship between immature regulators, $ u_{2}=u^{2}_{4} $, in the original variables:\\

$ a*X_{2}=bc^{-1}de^{-1}fg^{-1}i^{-1}jk^{-1}l*X^{2}_{4} $, or only $X_{2}=\tilde{K}*b*l* X^{2}_{4} $;\\

where $ b=(\beta_{2} + 1) $ and $l= (\beta_{4} + 1) $ important parameters considered in the next section, with $\tilde{K}  $ depending the other parameters and control modes $ \pi_{i}, \, \delta_{i} $. The normalised concentrations $ u_{2} $ and $ u_{4} $ always satisfies the condition in steady-state, this affine algebraic set, can be calculated easily simultaneously solving the parametric system equal to zero, i.e., $ \dot{X}_{1}=...=\dot{X}_{A}=\dot{X}_{R}=0 $, since the equivalence classes satisfied the equality.\\
In this point is important to mention, that the oscillatory genetic metabolic network can be completely described it by means of gluing maps which are toric morphisms of the dual cones into regular fan $ \sum= \cup_{i} \sigma^{\vee}_{i},\,i=1,2,3,4,A,R$. They are associated to monomial parametrizations of their respective differential equation in the S-system, for this genetic metabolic network  \textbf{(GMN)}.
\begin{center}
$  GMN=\left\lbrace \psi_{\sigma_{1},\sigma_{A}}, \psi_{\sigma_{A},\sigma_{2}}, \psi_{\sigma_{2},\sigma_{3}}, \psi_{\sigma_{3},\sigma_{R}},\psi_{\sigma_{R},\sigma_{4}}, \psi_{\sigma_{4},\sigma_{1}} \,\vert \, \sum =\cup_{i} \sigma^{\vee}_{i} , regular fan\right\rbrace  $ .
\end{center}
We claim that any genetic metabolic network described by this mathematical modelling (S-systems) can be represented by gluing maps or toric morphisms, and we write.\\

\begin{center}
$  GMN=\left\lbrace  \psi_{\sigma_{i},\sigma_{j}},  \, \vert \,\sigma_{i},\sigma_{j}\,\in\, \sum=\cup_{k=1}^{n} \sigma^{\vee}_{k} \right\rbrace $.
\end{center}

with $ \sum \, \subseteq \,\mathbb{R}^{n} $ regular fan, if the S-system consist of an irregular fan, then we apply the Hilbert basis technique. \\
Also following the central dogma of the molecular biology (which states that information only flows from DNA to RNA to protein), any genetic-metabolic system can be represented in a cascade network of fluxes of biomass (as shown above). This cascade of information is well represented geometrically and algebraically for the concatenation of gluing maps, as is shown below:
\begin{center}
$  \psi_{\sigma_{1},\sigma_{A}}\longrightarrow \psi_{\sigma_{A},\sigma_{2}}\longrightarrow \psi_{\sigma_{2},\sigma_{3}}\longrightarrow \psi_{\sigma_{3},\sigma_{R}} \longrightarrow \psi_{\sigma_{R},\sigma_{4}} \longrightarrow \psi_{\sigma_{4},\sigma_{1}}  $.
\end{center}
Each one of these maps contains algebraic information for re-parametrization of the binomials in the S-system, as we showed in the construction of the toric varieties.\\

\textbf{Obtaining a sustained oscillatory phenotypic region.}\\

In this section, we will obtain the complex region, studied by M. Savageau \cite{LomnitzSavageau2014}, which represents a sustained oscillatory phenotype. This phenotypic region is preserved under different control modes of the transcriptional regulation for this gene network.
The results presented in this work only include the steady-state solution of the system. Even though only we work with the dynamical solution of the regulator gene $ X_{R} $ which is suggested to intersect the toric ideals with algebraic set most common at all, in this case were the ideals $ <u_{A} - 1> $ and $ <u_{R} - 1> $, as is shown below.

We are interested in obtaining common phenotypic polytopes (phenotypic polytope). Intersecting the toric ideals gives the ideal most common to the complete gene circuit provided this intersection is not empty. A geometric interpretation is that we want common faces into the lattice polytope, these common faces are hyperplanes of concentrations that are sustained or conserved during all the steady-state available to the gene network. Also is important to say that this phenotypic polytope is the sustained oscillatory phenotype reproduced for the case where the solution has complex eigenvalues with positive real-part, shown by Savegau \cite{LomnitzSavageau2014}, and now we reproduce the result with toric ideals, the toric ideal associated to this oscillatory region is named as a \textbf{phenotypic toric ideal}.

Then the intersection of the ideals with Zariski closure not empty, yields:
\begin{center}
$ I_{P_{1}}=\langle \dot{X}_{A} \rangle \cap \langle \dot{X}_{1} \rangle \cap \langle \dot{X}_{2} \rangle \cap \langle \dot{X}_{3} \rangle = I_{P_{1}}=\langle {u}_{R} - 1 \rangle $,\\
$ I_{P_{2}}=\langle \dot{X}_{R} \rangle \cap \langle \dot{X}_{1} \rangle \cap \langle \dot{X}_{3} \rangle \cap \langle \dot{X}_{4} \rangle = I_{P_{2}}=\langle {u}_{A} - 1 \rangle $.
\end{center}
or in a geometric manner,
\begin{center}
$  \sigma^{\vee}_{P_{1}}=\sigma^{\vee}_{A} \cap \sigma^{\vee}_{2} \cap \sigma^{\vee}_{3} \cap \sigma^{\vee}_{R} \cap \sigma^{\vee}_{4}= $ Con $(e_{5})$,\\
$  \sigma^{\vee}_{P2}=\sigma^{\vee}_{R} \cap \sigma^{\vee}_{1} \cap \sigma^{\vee}_{3}  \cap \sigma^{\vee}_{4}= $ Con $(e_{6})$.
\end{center}
Taking the Zariski closure in both ideals, $Z(I_{P_{1}})=\lbrace u_{R} - 1=0 \rbrace   $ and $ Z(I_{P_{2}})=\lbrace u_{A} - 1=0 \rbrace  $, so we have .\\
\begin{center}
$ u_{A}= A*X_{A}=1$ hence we solve for $ X_{A}=A^{-1} $,\\

$ u_{R}=B*X_{R}=1 $, analogously for $ X_{R}=B^{-1} $.
\end{center}
With parameters $ A $ and $ B $ defined below; substituting in the equations with relationship to the repressor $ B*X_{R}=1 $ and activator $ A*X_{A}=1 $ respectively, in the equations,
\begin{center}
$ \dot{X}_{2} =A*X_{A}- (\beta_{2} + \mu)*X_{2}=1-(\beta_{2} + \mu)*X_{2}$,\\
$ \dot{X}_{4} =B*X_{R}- (\beta_{4} + \mu)*X_{4}=1-(\beta_{4} + \mu)*X_{4}$.
\end{center}

And this is a system of first order differential equations with solutions well known, in the literature, and given by:
\begin{center}
$  X_{2}(t)=X_{20}*\exp (-(\beta_{2} + \mu)*t) +(\beta_{2} + \mu)^{-1} \left(1 -C_{0} \exp (-(\beta_{2} + \mu)*t) \right) $,\\

$  X_{4}(t)=X_{40}*\exp (-(\beta_{4} + \mu)*t) +(\beta_{4} + \mu)^{-1} \left(1 -\tilde{C}_{0} \exp (-(\beta_{4} + \mu)*t) \right)  $.
\end{center}
With real constants $ X_{20} $ and $ X_{40} $ which are the initial conditions of concentration of mature regulators $ X_{2} $ and $ X_{4} $ respectively, for the solution of the homogeneous differential equation.\\

\begin{comment}
And from the condition of Zariski closure, we have $ u_{A}=u_{R}=1 $, and the condition of steady state, from there it follows the equalities:
\begin{center}
$(\beta_{2} + \mu)X_{2}(0)= (\beta_{4} + \mu) X_{4}(0) $.
\end{center}
and then;
 \begin{center}

$(\beta_{2} + \mu) \left\lbrace  X_{20} +(\beta_{2} + \mu)^{-1} \left( 1 -C_{0} \right) \right\rbrace =  $\\
 $ (\beta_{4} + \mu)\left\lbrace  X_{40}  + (\beta_{4} + \mu)^{-1} \left(1 -\tilde{C}_{0} \right)  \right\rbrace $.
 \end{center}

Applying terms to both sides of the equality and ordering parameters, always that the initial production of molar concentrations in the time $ t_{0}= 0$ be zero, $ C_{0}=\tilde{C}_{0} $, and so is satisfied the analytical expression.
 \begin{center}
$  \log_{10}  \left\lbrace  X_{20}(\beta_{2} + \mu)  \right\rbrace = \log_{10} \left\lbrace X_{40}(\beta_{4} + \mu) \right\rbrace $.

\end{center}
And so we have the analytical expression for the polytope phenotypic, which is represented in complex plane in the yellow regions in the figures reported above as is shown.
\begin{center}
$\log_{10}(\beta_{4}/  \mu + 1) =\log_{10} (\beta_{2} / \mu + 1  ) + \log_{10}(X_{20}/X_{40})  $.
\end{center}

Where the  $ I_{P} $ is the \textbf{phenotypic toric ideal} and $\sigma^{\vee}_{P}  $ is a \textbf{phenotypic polytope} which is a generator ray in the direction of $ e_{5} $. Taking the Zariski closure of this ideal, $ Z(I_{P})=\lbrace u_A} - 1=0 \rbrace $ are the common solutions to all gene regulatory dynamics of the network.
\end{comment}

Also taking the Zariski closure for both ideals $ I_{\sigma^{\vee}_{2}} $ and $I_{\sigma^{\vee}_{4}} $ we have.
\begin{center}
 $Z((I_{\sigma^{\vee}_{2}}))=\left\lbrace  u_{2}u_{A}-1=0, u_{1}-1=0, u_{R}-1=0, u_{3}- 1=0, u_{4}- 1=0   \right\rbrace $.\\
$Z((I_{\sigma^{\vee}_{4}}))=\left\lbrace  u_{4}u_{R}-1=0, u_{1}-1=0, u_{2}-1=0, u_{3}- 1=0, u_{A}- 1=0   \right\rbrace $.

\end{center}
To reproduce the oscillatory phenotype of Savageau, now we study the non-trivial generators, in both Zariski closures above, where to study these kind of solutions associated to Zariski closure, we are understanding, that is a equivalent way to study the steady-state condition associated to each pair $ (Z((I_{\sigma^{\vee}_{i}})),\dot{u}_{i} ) $ for all $ i=1,2,3,4,A,R $; i.e, in our case we want to reproduce the oscillatory phenotype related to the regulators $ u_{2}=(\beta_{2} + \mu)X_{2} $ and $ u_{4}= (\beta_{4} + \mu)X_{4} $ in normalised coordinates. \\
Taking the non-trivial generators above, we have $u_{2}u_{A}-1=0  $ and $  u_{4}u_{R}-1=0$, we can appreciate the inverse relationship between immature and mature regulators, $ u_{A}=u^{-1}_{2} $ and $ u_{R}=u^{-1}_{4}  $, which tell us, most production of mature regulators, less concentration of immature regulators. And substituting in steady-state from the analytical solution given above for $ X_{2}(0) $ and $ X_{4}(0) $. we have $ u_{A}=(\beta_{2} + \mu)^{-1}X^{-1}_{20} $ and $ u_{A}=(\beta_{4} + \mu)^{-1}X^{-1}_{40} $, we substitute in the expression for steady-state equations, $ \dot{X}_{2}=\dot{X}_{4}=0 $, therefore,\\

\begin{center}
$ 0=\dot{X}_{2} = (\beta^{0}_{2} + \mu^{0}) \left( \dfrac{\beta_{A}}{\beta^{0}_{A}}\dfrac{\alpha_{1}^{0} n_{2}^{0}}{(\beta_{1}^{0} + \mu^{0})\sqrt{\rho_{1}^{0}} K_{2}^{0}} \right) *(\beta_{2} + \mu)^{-1}X^{-1}_{20} - (\beta_{2} + \mu)*X_{20} $,\\

$ 0=\dot{X}_{4}=(\beta^{0}_{4} + \mu^{0}) \left( \dfrac{\beta_{R}}{\beta^{0}_{R}}\dfrac{\alpha_{3}^{0} n_{4}^{0}}{(\beta_{3}^{0} + \mu^{0})\sqrt{\rho_{3}^{0}} K_{4}^{0}} \right) *(\beta_{4} + \mu)^{-1}X^{-1}_{40} - (\beta_{4} + \mu)X_{40} $.

\end{center}
\begin{center}

\end{center}
This type of solutions is suggested by M. Savageau \cite{LomnitzSavageau2014}. Now, we have the desired relation between the regulatory capacities: $ (\beta_{2} + \mu) $ and $( \beta_{4}+ \mu) $. Substituting and clearing terms:
\begin{center}
$ (\beta_{4} + \mu)^{2}*X^{2}_{40}(\beta_{3}^{0} + \mu^{0})\beta^{0}_{R}\sqrt{\rho_{3}^{0}} K_{4}^{0} K_{4A}K_{2}^{0}(\beta^{0}_{2} + \mu^{0})\beta_{A}\alpha_{1}^{0} n_{2}^{0}*Z_{1}= (\beta_{2} + \mu)^{2}*X^{2}_{20}*(\beta_{1}^{0} + \mu^{0}) \sqrt{\rho_{1}^{0}} K_{2}^{0} K_{2} K_{4}^{0} \beta_{R} \alpha_{3}^{0} n_{4}^{0} (\beta^{0}_{4} + \mu^{0}) *Z_{2}$.
\end{center}
clearing $( \beta_{4}+ \mu) $ in the left side and $ (\beta_{2} + \mu) $ in the right side, and after applying logarithms in both sides, we have the expression:

\begin{center}
$ \log_{10} \left( \beta_{4}/ \mu+ 1 \right) =\log_{10} \left\lbrace  \dfrac{X_{40}}{X_{20}} \sqrt{\tilde{K}} \left(  \beta_{2}/ \mu + 1 \right) \right\rbrace  = \log_{10}( \beta_{2}/ \mu + 1 ) + \log_{10} \left(  \bar{Z} \sqrt{\tilde{K}}\right)  $.
\end{center}
This geometrical region can be seen in Figures 3-9; \cite{LomnitzSavageau2014}, given that there is free selection of some parameters (ref. \cite{LomnitzSavageau2014}), the regions change with respect to them; $ \mu, \, \beta_{A}, \, \beta^{0}_{A}, \, \beta_{R}, \, \beta^{0}_{R} $ are free parameters without experimental measure  and the complex number $ \bar{Z}=\dfrac{X_{40}}{X_{20}}=R\exp(\imath*\Theta) \,\in \, \mathbb{C}^{*}$ is given for periodical initial conditions of production regulators, M. Savageau \cite{LomnitzSavageau2014}, here $\tilde{K} $ is defined by:\\\\
$ \tilde{K}=\left( \dfrac{B}{A} \right) \left( \dfrac{K_{2R}}{K^{0}_{2}} \right) \left( \dfrac{K^{0}_{4}}{K_{4A}} \right)  \left(  \dfrac{ \beta^{0}_{4} + \mu^{0} }{ \beta^{0}_{2} + \mu^{0} } \right)  $, where;\\\\

$ A= \left[ \left( \dfrac{\beta_{A}}{\beta^{0}_{A}} \right)    \dfrac{\alpha_{1}^{0} n_{2}^{0}}{(\beta_{1}^{0} + \mu^{0})\sqrt{\rho_{1}^{0}}K_{2}^{0}} \right] $,\\\\

$ B=  \left[ \left( \dfrac{\beta_{R}}{\beta^{0}_{R}} \right) \dfrac{\alpha_{3}^{0} n_{4}^{0}}{(\beta_{3}^{0} + \mu^{0})\sqrt{\rho_{3}^{0}} K_{4}^{0}} \right]$.\\\\

Substituting some parameters for empirical values:, for details, see reference, J. Lomintz and M. Savageau \cite{LomnitzSavageau2014}. We have:\\\\

$ A= \left[ \left( \dfrac{\beta_{A}}{\beta^{0}_{A}} \right)    \dfrac{33.3 mRNAs (30)}{\sqrt{100}(1.5811)} \right] $,\\\\

$B=  \left[ \left( \dfrac{\beta_{R}}{\beta^{0}_{R}} \right) \dfrac{33.3 mRNAs(30)}{\sqrt{100} (0.5)} \right]$.\\\\
$ \dfrac{\beta_{A}}{\beta^{0}_{A}}=0.25 $ and $  \dfrac{\beta_{R}}{\beta^{0}_{R}}=0.175 $ (free values), see \cite{LomnitzSavageau2015}, \cite{LomnitzSavageau2014}, $ K^{0}_{2} \approx \sqrt{10} K^{0}_{4} $, $ K^{0}_{4} \ll 17.3938 $. Every pair of values $ K_{2R} $ and $ K_{4A} $ are calculated with respect to $ \pi_{i}=1, 0 $, $ \delta_{i}=1,0 $ and the formulas given at the begin of the example; also these values and formulas have a bit different normalization, but the intention here is to show that using common toric ideals in the S-system, it is possible to find an invariant region in this case the use of toric ideals, as was the case for $ \langle {u}_{A} - 1 \rangle  $ and $ \langle {u}_{R} - 1 \rangle $.\\\\

In the figures presented (Figures 3 -9), the rigth panel shows yellow geometric regions that reproduce the sustained oscilatory region found for different control modes, which are given by the condition lattice vector $ (\delta_{1},\delta_{3},\pi_{1},\pi_{3}) \,\in \, \mathbb{Z}^{2} $; shown in, J. Lomintz and M. Savageau \cite{LomnitzSavageau2014}. \\
Now we consider other analytical solutions for other phenotypic polytopes, as is the case associated to the phenotypic toric variety $ X_{\sum} =\left\lbrace u_{2}=u^{2}_{4} \right\rbrace$ where this affine space associated to the steady-state solution in all the S-System, and is possible to calculate new analytical expression from this expression, as is shown to continuation, we consider the condition of steady-state is always satisfied:
\begin{center}
$ X_{2}(0)=\tilde{K}*(\beta_{2}+ \mu)*(\beta_{4}+ \mu)X_{4}^{2}(0)  $,
\end{center}
And finally we obtain in logarithmic-space an analytical function defined in a branch of the complex logarithm, i.e., $ \log_{10}(\beta_{4}/\mu) :\mathbb{R}^{+} \longrightarrow \mathbb{C}$. where,
\begin{center}
$ \log_{10}(\beta_{4} / \mu + 1)= \log_{10} \left\lbrace \dfrac{X_{20}}{\tilde{K}*\mu^{2}*X_{4}^{2}(0)}\right\rbrace  - \log_{10}(\beta_{4} / \mu + 1)  $.
\end{center}
In its polar form, it is the case when the value of $ \tilde{\theta} $ has complex values, and in this way we have associated the sustained oscillatory complex region:
\begin{center}
$ \log_{10}(\beta_{4}/\mu + 1)= \log_{10}\left[\tilde{R} \left( \cos(\tilde{\theta}) \pm \imath \sin(\tilde{\theta})  \right) \right]  $.
\end{center}
Where the parameter $ \tilde{K} $ depend of all control modes of oscillatory gene circuit studied, $ \pi^{i}=\delta_{i}=1,0 $ with combinations given in the figures 3-9 for $i=1,2,3,4,A,R $ and  $ \tilde{K} $ is varying in these numbers, each region is visualised in the respective figures. Also the parameters $ \tilde{R} $ and $\tilde{\theta} $ are functions of the parametric variable $(\beta_{2}/\mu + 1)$.

Here we consider constants initial conditions $X_{2}(0)  $ and  $ X_{4}^{2}(0) $ from solution of differential equations, and also to recover the sustained oscillatory region obtained for Savageau we give nominal values for its graphical representation.\\

This expression to point us what an oscillatory sustained region be preserved for, a possible case of biological interpretation, could be that the initial concentration $ C_{0} $ of mature regulator $ X_{2} $ must be constant, as is the case of $  u_{A}=1 $ where was associated the non homogeneous solution of the ordinary differential equation to $ \dot{X}_{2} $, with initial condition $C_{0}$, and the initial concentration of immature regular $ X_{20} $, must be null or almost zero, and $1 \approx \, C_{0}$ must be true.\\

\begin{comment}

\begin{center}
$(\beta_{2}+ \mu)\left\lbrace  X_{20} + (\beta_{2}+ \mu)^{-1}(1-C_{0})\right\rbrace =(\beta_{4}+ \mu)^{2} \left\lbrace  X_{40} + (\beta_{4}+ \mu)^{-1}(1-\tilde{C}_{0})\right\rbrace^{2}  $,
\end{center}
clearing and grouping terms, we have the expression:\\
$(\beta_{4}+ \mu)^{2}(X_{40})^{2}+ 2X_{40}(1- \tilde{C}_{0})(\beta_{4}+ \mu)^{2}(X_{40}) + \left\lbrace (1- \tilde{C}_{0})^{2} - (1- \tilde{C}_{0}) - X_{20}(\beta_{2}+ \mu) \right\rbrace =0 $.\\
From here, it follows:
\begin{center}
$\tilde{A} (\beta_{4}+ \mu)^{2} + \tilde{B} (\beta_{4}+ \mu) + \tilde{C} =0$.
\end{center}
with parameters $ \tilde{A},\, \tilde{B}, \,  \tilde{C} $ defined as:\\
$\tilde{A}=(X_{40})^{2}$,\\
$\tilde{B}= 2X_{40}(1- \tilde{C}_{0}) $ ,\\
$\tilde{C} =\left\lbrace (1- \tilde{C}_{0})^{2} - (1- \tilde{C}_{0}) - X_{20}(\beta_{2}+ \mu) \right\rbrace $.

So obtaining an algebraic equation of second degree, where we solve it for the parametric variable $ (\beta_{4}+ \mu) $ and its solution will depend of the parametric variable one $ (\beta_{2}+ \mu) $, they are the same that generate the complex phenotypic region in Savageau; but their analytical expression is different, so there exists a new complex phenotypic region, invariant during the phase in steady-state at all oscillatory genetic network, using the general formula, we have:
$(\beta_{4}+ \mu)=\dfrac{-\tilde{B} \pm \sqrt{(\tilde{B})^{2}-4 \tilde{A}\tilde{C}}}{2\tilde{A}}  $.
Simplifying and substituting the parametric variables $ \tilde{A},\, \tilde{B}, \,  \tilde{C} $.\\
\begin{center}
$  \beta_{4}/\mu= \dfrac{-(1- \tilde{C}_{0})}{X_{40} \mu} \pm \dfrac{1}{X_{40}\mu} \left( \sqrt{X_{20}(\beta_{2}+ \mu) +1 -C_{0}} \right)  -1$.
\end{center}
And finally we obtain in logarithmic-space an analytical function defined in a branch of the complex logarithm, i.e., $ \log_{10}(\beta_{4}/\mu) :\mathbb{R}^{+} \longrightarrow \mathbb{C}$. where,
\begin{center}
$ \log_{10}(\beta_{4}/\mu)= \log_{10}\left[\dfrac{-(1- \tilde{C}_{0})}{X_{40} \mu} \pm \dfrac{1}{X_{40}\mu} \left( \sqrt{X_{20}(\beta_{2}+ \mu) +1 -C_{0}} \right)  -1   \right]  $.
\end{center}
In its polar form:
\begin{center}
$ \log_{10}(\beta_{4}/\mu)= \log_{10}\left[\left( \tilde{R}\cos(\tilde{\theta}) \pm \imath \sin(\tilde{\theta})  \right) -1   \right]  $.
\end{center}
Is easy to see, that the amplitude $\tilde{R}$ and frequency $\tilde{\theta}  $ of the signal, depend of the parametric values $ \tilde{A},\, \tilde{B}, \,  \tilde{C} $ anew, where $ \imath $ is the imaginary unit. \\

The complex region is preserved, if only if the following inequality is satisfied,
\begin{center}
$ X_{20}(\beta_{2}+ \mu) +1 < \, C_{0} $.
\end{center}
This expression to point us what an oscillatory sustained region be preserved for, a possible case of biological interpretation, could be that the initial concentration $ C_{0} $ of mature regulator $ X_{2} $ must be constant, as is the case of $  u_{A}=1 $ where was associated the non homogeneous solution of the ordinary differential equation to $ \dot{X}_{2} $, with initial condition $C_{0}$, and the initial concentration of immature regular $ X_{20} $, must be null or almost zero, and $1 \approx \, C_{0}$ must be true, there are many other cases of the inequality that can preserve the oscillatory region.\\
\end{comment}

A lattice diagram of toric ideals as subgroups can be done; where the generators $ <u_{A}-1> $ and $ <u_{R}- 1> $ associated to the oscillatory phenotype found in Savageau can be localized in such diagram, respect to a partial order, for exmaplle inclusion of sets. The rest of generators are not yet explored and they also were found  with Hilbert basis as realization of a toric variety $ X_{\sum} $, and will be explored in detail in a future analysis.

\section{Comments and future developments.}
The mathematical results here presented will enable  the study of S-Systems in steady state by means of computing toric ideals, since minimal generators contain the solution space of S-Systems. In this way the use of Hilbert basis is important for the solution of these systems, even more with Theorem 4.5 which gives a clear perspective for the study of dynamical solutions on genetic-metabolic networks at  hand with Hilbert basis associated to the lattice polytope in S-Systems. These will enable us in a future work,  to model large genetic- metabolic networks.
\section{Acknowledgment}
We acknowledge funding from UNAM and from the National Institutes of Health (grant number 5R01GM110597-03). Also we give the thanks for her valuable comments, and appreciate all the help and assistance proportioned by the Dr. Margareta Boege for the improvement of this work.

\section{Conclusions.}

The main result in this work is the use of \textbf{Theorem 3.6}, and the \textbf{Hironaka-Atiyah's Theorem} for the demonstrations given in Theorems, Theorem 4.4-(\textbf{embedding of S-Systems in toric varieties}), and Theorem 4.5-(\textbf{toric resolution on S-Systems}), that give the foundation to the toric formalization of genetic-metabolic networks in the approach studied by M.A. Savageau. These facts demonstrate that S-Systems are toric varieties. As a consequence of these results, the Theorem 4.4 give us a formalization for the study of S-Systems in steady-state. In more general terms, the Theorem 4.5 provides the formal support to the use of toric algebraic geometry tools to the study of the dynamics of genetic-metabolic networks. In both cases we have seen the use of \textit{Hilbert basis} for the realizations of toric morphisms, and computationally it opens the possibility for this mathematical modelling of large-scale genetic-metabolic networks, since the algorithms to compute Hilbert basis in high-dimensional lattice polytope can be used. Also we analyse a particular genetic-metabolic network, computing the toric ideals and toric varieties associated to the S-system of this network, this network is the case of a synthetic oscillating design, studied by Savageau \cite{LomnitzSavageau2014} in the which was reported the existence of a sustained oscillating region (oscillating phenotype) when the regulation  transcriptional was submitted under different control modes, we reproduced this region with an ideal toric common called phenotypic toric ideal, and we also constructed the phenotypic toric variety, where we can conclude that these geometrical-algebraic objects are invariant too under these control modes, finding the possibility of more intrinsic properties in the topology of any gene-metabolic network, and to perform future studies of these structures.

\input amsart-template.bbl

\end{document}