Computing critical points for algebraic systems defined by hyperoctahedral invariant polynomials

TL;DR

This paper develops an algorithm to compute critical points of algebraic systems invariant under the hyperoctahedral group, leveraging symmetry to efficiently describe solution sets with polynomial runtime.

Contribution

It introduces hyperoctahedral representations for invariant sets and presents an algorithm exploiting symmetry to compute critical points efficiently.

Findings

Algorithm has polynomial runtime in the size of the output.

Uses symmetry to reduce computational complexity.

Provides a new representation for invariant solution sets.

Abstract

Let $\mathbb{K}$ be a field of characteristic zero and $\mathbb{K}[x_1, \dots, x_n]$ the corresponding multivariate polynomial ring. Given a sequence of $s$ polynomials $\mathbf{f} = (f_1, \dots, f_s)$ and a polynomial $\phi$, all in $\mathbb{K}[x_1, \dots, x_n]$ with $s<n$, we consider the problem of computing the set $W(\phi, \mathbf{f})$ of points at which $\mathbf{f}$ vanishes and the Jacobian matrix of $\mathbf{f}, \phi$ with respect to $x_1, \dots, x_n$ does not have full rank. This problem plays an essential role in many application areas. In this paper we focus on a case where the polynomials are all invariant under the action of the signed symmetric group $B_n$. We introduce a notion called {\em hyperoctahedral representation} to describe $B_n$-invariant sets. We study the invariance properties of the input polynomials to split $W(\phi, \mathbf{f})$ according to the orbits of $B_n$ and then design an algorithm whose output is a {hyperoctahedral representation} of $W(\phi, \mathbf{f})$. The runtime of our algorithm is polynomial in the total number of points described by the output.