Connectivity in Symmetric Semi-Algebraic Sets
Cordian Riener, Robin Schabert, Thi Xuan Vu
TL;DR
This work addresses deciding equivariant connectivity for semi-algebraic sets defined by symmetric polynomials of degree at most $d\le n$, assuming the two query points lie in the same Weyl chamber after sorting. The authors exploit the $S_n$-action via Vandermonde varieties and a contraction to the $d$-dimensional orbit boundary, reducing global connectivity to boundary connectivity computable with a graph-based framework. The core contribution is a polynomial-time algorithm (for fixed $d$ and a constant number of polynomials) that constructs a connectivity graph over $d$-dimensional faces and uses a Vandermonde-based minimization subroutine (MV) to connect inputs through the graph, with key subroutines costing $s^{d+1}d^{O(d^2)}$ arithmetic operations. This advances roadmap-style connectivity in symmetric settings and opens paths to broader symmetric semi-algebraic connectivity and homology computations.
Abstract
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 $n$.