arXiv subjects

Robin Schabert

Publications and source records attributed to Robin Schabert.

7 recordsLinked to original sources

arXiv2026

Any-dimensional Positivstellens\"atze for symmetric functions

Positivstellens\"atze provide certificates of positivity for polynomials. Extending these certificates to symmetric functions, uniformly across all dimensions, presents structural challenges. For instance, the underlying domain is not semialgebraic. In this paper, we prove two Positivstellens\"atze for symmetric functions that are uniformly bounded below by some $\varepsilon > 0$. These are infinite-dimensional analogs of theorems of P\'olya and Reznick. The proof relates evaluations of the (truncated) power sum map $(p_2,p_3,\dots)$ to moments of discrete probability measures on the compact interval $[-1,1]$. This yields a characterization of the closure of the orbit space of the infinite symmetric group on the sphere. Finally, we provide an alternative proof of existing Positivstellens\"atze for normalized symmetric functions.

math.AG↗

arXiv2025

Deciding Connectivity in Symmetric Semi-Algebraic Sets

A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same connected component. We focus on the case where the defining equations and inequalities are invariant under the natural action of the symmetric group and where each polynomial has degree at most $ d $, with $ d < n $ (where $ n $ denotes the number of variables). Exploiting this symmetry, we develop and analyze algorithms for two key tasks. First, we present an algorithm that determines whether the orbits of two given points are connected. Second, we provide an algorithm that decides connectivity between arbitrary points in the set. Both algorithms run in polynomial time with respect to $ n $.

cs.SC↗

arXiv2024

Constructively describing orbit spaces of finite groups by few inequalities

Let $G$ be a finite group acting linearly on $\mathbb{R}^n$. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space $\mathbb{R}^n /\!/G$ as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Br\"ocker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group $G$. Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Br\"ocker concerning the genericity of his result.

math.AG↗

arXiv2024

Connectivity in Symmetric Semi-Algebraic Sets

Semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case when all equations and inequalities are invariant under the action of the symmetric group and their degrees at most $d<n$, where $n$ is the number of variables. Additionally, we assume that the two points are in the same fundamental domain of the action of the symmetric group, by assuming that the coordinates of two given points are sorted in non-decreasing order. We construct and analyze an algorithm that solves this problem, by taking advantage of the group action, and has a complexity being polynomial in $n$.

cs.SC↗

arXiv2024

Shellable slices of hyperbolic polynomials and the degree principle

We study a natural stratification of certain affine slices of univariate hyperbolic polynomials. We look into which posets of strata can be realized and show that the dual of the poset of strata is a shellable simplicial complex and in particular a combinatorial sphere. From this we obtain a g-theorem and an upper bound theorem on the number of strata. We use these results to design smaller test sets to improve upon Timofte's degree principle and give bounds on how much the degree principle can be improved.

math.AG↗

arXiv2024

Slices of Stable Polynomials and Connections to the Grace-Walsh-Szeg\H{o} theorem

Univariate polynomials are called stable with respect to a domain $D$ if all of their roots lie in $D$. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in $D^n$ with few distinct coordinates is necessary and sufficient for the intersection with $D^n$ to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szeg\H{o}'s coincidence theorem.

math.AG↗

arXiv2022

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article are families of hyperbolic polynomials which are determined through $k$ linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a \emph{hyperbolic slice}. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the \emph{local extreme points} of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most $k$ distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

math.AG↗