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 (2003) [15]. It says that a symmetric real polynomial F of degree d in n variables is positive on R^n if and only if it is non-negative on the subset of points with at most max{⌊d/2⌋,2} distinct components. We deduce Timofte’s original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea that we are using to prove this statement is that of relating 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.