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Deciding Connectivity in Symmetric Semi-Algebraic Sets

Cordian. Riener, Robin Schabert, Thi Xuan Vu

TL;DR

This work addresses the problem of deciding connectivity in symmetric semi-algebraic sets invariant under the action of the symmetric group $S_n$, under the condition that the defining polynomials have degree $d<n$. It exploits a dimension-reduction strategy via symmetric polynomials, Weyl chambers, and Vandermonde maps to relate full-space connectivity to lower-dimensional orbit boundaries and to orbit connectivity. The authors present two polynomial-time algorithms: one for testing connectivity between $S_n$-orbits and another for connectivity between arbitrary points, with complexities that are polynomial in $n$ for fixed $d$, alongside a graph-based Mayer–Vietoris framework and a mirrored-space (Coxeter) extension to handle walls and chamber transitions. The results offer practical, structure-exploiting tools for connectivity questions in symmetric real algebraic sets and open avenues for broader Coxeter-group generalizations and representations beyond power sums. The approach has potential impact on real-algebraic geometry, invariant theory, and applications where symmetry reduces problem size while preserving topological queries.

Abstract

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 \( n \).

Deciding Connectivity in Symmetric Semi-Algebraic Sets

TL;DR

This work addresses the problem of deciding connectivity in symmetric semi-algebraic sets invariant under the action of the symmetric group , under the condition that the defining polynomials have degree . It exploits a dimension-reduction strategy via symmetric polynomials, Weyl chambers, and Vandermonde maps to relate full-space connectivity to lower-dimensional orbit boundaries and to orbit connectivity. The authors present two polynomial-time algorithms: one for testing connectivity between -orbits and another for connectivity between arbitrary points, with complexities that are polynomial in for fixed , alongside a graph-based Mayer–Vietoris framework and a mirrored-space (Coxeter) extension to handle walls and chamber transitions. The results offer practical, structure-exploiting tools for connectivity questions in symmetric real algebraic sets and open avenues for broader Coxeter-group generalizations and representations beyond power sums. The approach has potential impact on real-algebraic geometry, invariant theory, and applications where symmetry reduces problem size while preserving topological queries.

Abstract

A semi-algebraic set is a subset of 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 , with (where 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 .

Paper Structure

This paper contains 19 sections, 19 theorems, 55 equations, 6 algorithms.

Table of Contents

  1. Introduction
  2. Prior Results
  3. Our main results
  4. Relation to a previous version
  5. Algorithmic preliminaries
  6. Encoding of real algebraic points
  7. Roadmaps and connectivity
  8. Preliminaries on symmetric polynomials
  9. Symmetric polynomials
  10. Weyl chambers and restriction to faces
  11. Vandermonde maps
  12. Graphs and connectivity of unions
  13. Connectivity in the orbit space
  14. Restriction to subspaces
  15. Algorithm for testing the connectivity within one Weyl chamber
  16. ...and 4 more sections

Key Result

Theorem 2.4

Let $F$ be a sequence of polynomials in $\mathbb{Q}[\mathbf{X}]$ with the algebraic set $Z \subset \mathbb{R}^n$ defined by $F$ of dimension $k$. Consider $G = (g_1, \dots, g_s) \subset \mathbb{K}[\mathbf{X}]$. Let $S$ be a semi-algebraic subset of $Z$ defined by $G$. Let $d$ be a bound for the degr As a consequence, there exists an algorithm for deciding whether two given points, described by rea

Theorems & Definitions (52)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Theorem 2.4: basu2000computing and BPR06
  • Remark 3.1
  • Definition 3.2
  • Proposition 3.3
  • Definition 3.4
  • Example 3.5
  • Theorem 3.6
  • ...and 42 more