Test Sets for Nonnegativity of Polynomials Invariant under a Finite Reflection Group

TL;DR

This paper establishes hyperplane unions as nonnegativity witnesses for invariant polynomials under reflection groups, providing sharp bounds and characterizations, and proposes a conjecture extending Timofte's half-degree principle.

Contribution

It introduces hyperplane unions as nonnegativity witnesses for invariant polynomials, with sharp degree bounds and characterizations, and extends Timofte's principle to reflection groups.

Findings

Hyperplanes perpendicular to root systems serve as nonnegativity witnesses.

Degree bounds for invariance are sharp for groups with multiplication by -1.

A conjecture generalizes Timofte's half-degree principle for reflection groups.

Abstract

A set $S\subset \mathbb{R}^n$ is a nonnegativity witness for a set $U$ of real homogeneous polynomials if $F$ in $U$ is nonnegative on $\mathbb{R}^n$ if and only if it is nonnegative at all points of $S$. We prove that the union of the hyperplanes perpendicular to the elements of a root system $\Phi\subseteq \mathbb{R}^n$ is a witness set for nonnegativity of forms of low degree which are invariant under the reflection group defined by $\Phi$. We prove that our bound for the degree is sharp for all reflection groups which contain multiplication by $-1$. We then characterize subspaces of forms of arbitrarily high degree where this union of hyperplanes is a nonnegativity witness set. Finally we propose a conjectural generalization of Timofte's half-degree principle for finite reflection groups.