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 $n$.