On the degree and half degree principle for symmetric polynomials

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 ${\mathbb{R}}^n$ (on ${\mathbb{R}}_{\ge 0}^n$) 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.

(v2) revision based on suggestions by referee

References in corpus (1)

• Polya-Schur master theorems for circular domains and their boundaries

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