Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Universality Theorem for Nested Polytopes   In a nutshell, we show that polynomials and nested polytopes are topological,
algebraic and algorithmically equivalent. Given two polytops $A\subseteq B$ and
a number $k$, the Nested Polytope Problem (NPP) asks, if there exists a
polytope $X$ on $k$ vertices such that $A\subseteq X \subseteq B$. The polytope
$A$ is given by a set of vertices and the polytope $B$ is given by the defining
hyperplanes. We show a universality theorem for NPP. Given an instance $I$ of
the NPP, we define the solutions set of $I$ as $$ V'(I) = \{(x_1,\ldots,x_k)\in
\mathbb{R}^{k\cdot n} : A\subseteq \text{conv}(x_1,\ldots,x_k) \subseteq B\}.$$
As there are many symmetries, induced by permutations of the vertices, we will
consider the \emph{normalized} solution space $V(I)$. Let $F$ be a finite set
of polynomials, with bounded solution space. Then there is an instance $I$ of
the NPP, which has a rationally-equivalent normalized solution space $V(I)$.
Two sets $V$ and $W$ are rationally equivalent if there exists a homeomorphism
$f : V \rightarrow W$ such that both $f$ and $f^{-1}$ are given by rational
functions. A function $f:V\rightarrow W$ is a homeomorphism, if it is
continuous, invertible and its inverse is continuous as well. As a corollary,
we show that NPP is $\exists \mathbb{R}$-complete. This implies that unless
$\exists \mathbb{R} =$ NP, the NPP is not contained in the complexity class NP.
Note that those results already follow from a recent paper by Shitov. Our proof
is geometric and arguably easier.

6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
30
The categories of this paper are: q-alg
16
The categories of this paper are: stat
37
The categories of this paper are: supr-con
10
The categories of this paper are: q-bio
23
The categories of this paper are: solv-int
13
The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
35
The categories of this paper are: mtrl-th
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an