Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Quadratization of ODEs: Monomial vs. Non-Monomial   Quadratization is a transform of a system of ODEs with polynomial right-hand
side into a system of ODEs with at most quadratic right-hand side via the
introduction of new variables. It has been recently used as a pre-processing
step for new model order reduction methods, so it is important to keep the
number of new variables small. Several algorithms have been designed to search
for a quadratization with the new variables being monomials in the original
variables. To understand the limitations and potential ways of improving such
algorithms, we study the following question: can quadratizations with not
necessarily monomial new variables produce a model of substantially smaller
dimension than quadratization with only monomial new variables?
  To do this, we restrict our attention to scalar polynomial ODEs. Our first
result is that a scalar polynomial ODE
$\dot{x}=p(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots + a_0$ with $n\geqslant 5$ and
$a_n\neq0$ can be quadratized using exactly one new variable if and only if
$p(x-\frac{a_{n-1}}{n\cdot a_n})=a_nx^n+ax^2+bx$ for some $a, b \in
\mathbb{C}$. In fact, the new variable can be taken
$z:=(x-\frac{a_{n-1}}{n\cdot a_n})^{n-1}$. Our second result is that two
non-monomial new variables are enough to quadratize all degree $6$ scalar
polynomial ODEs. Based on these results, we observe that a quadratization with
not necessarily monomial new variables can be much smaller than a monomial
quadratization even for scalar ODEs.
  The main results of the paper have been discovered using computational
methods of applied nonlinear algebra (Gr\"obner bases), and we describe these
computations.

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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: alg-geom
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The categories of this paper are: nucl-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: q-alg
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The categories of this paper are: nlin
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: hep-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: econ
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The categories of this paper are: acc-phys