On the degree and half degree principle for symmetric polynomials

TL;DR

This paper provides an elementary proof of a criterion for the positivity of symmetric polynomials based on the number of distinct components, connecting it to linear optimization in orbit spaces.

Contribution

It introduces a new, elementary proof of Timofte's degree and half-degree principles for symmetric polynomials, with potential applicability to other groups.

Findings

The positivity of symmetric polynomials can be checked on a small subset of points.

The proof relates polynomial positivity to linear optimization in orbit spaces.

Elementary methods simplify understanding of symmetric polynomial positivity.

Abstract

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it is so on the subset of points with at most $\max\{\lfloor d/2\rfloor,2\}$ distinct components. We deduce Timofte's original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea we are using to prove this statement is to relate it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group $S_n$ this can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.