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 \( n \).

翻译：半代数集合是由有限个多项式方程和不等式定义的 $\mathbb{R}^n$ 子集。本文研究判定该集合中两个点是否属于同一连通分支的问题。我们重点关注定义方程与不等式在对称群自然作用下保持不变，且每个多项式次数至多为 \( d \)（其中 \( d < n \)，\( n \) 表示变量个数）的情形。利用该对称性，我们开发并分析了针对两个关键任务的算法。首先，我们提出一种判定两个给定点轨道是否连通的算法。其次，我们给出一种判定集合中任意两点间连通性的算法。两种算法关于 \( n \) 均具有多项式时间复杂度。