Deciding positivity of multisymmetric polynomials

TL;DR

This paper explores non-negativity certification of multisymmetric polynomials, providing a polynomial-time method for testing convexity in this class, contrasting with the NP-hardness in the general case.

Contribution

It generalizes symmetric polynomial non-negativity characterization to multisymmetric polynomials and offers a polynomial-time convexity test for fixed-degree multisymmetric polynomials.

Findings

Certification of non-negativity for multisymmetric polynomials

Polynomial-time convexity testing for fixed-degree multisymmetric polynomials

Contrast with NP-hardness in general polynomial convexity testing

Abstract

The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of multisymmetric polynomials. In this setting we generalize the characterization of non-negative symmetric polynomials by adapting the method of proof developed by the second author. One particular case where our results can be applied is the question of certifying that a (multi-)symmetric polynomial defines a convex function. As a direct corollary of our main result we are able to derive that in the case of (multi-)symmetric polynomials of a fixed degree testing for convexity can be done in a time which is polynomial in the number of variables. This is in sharp contrast to the general case, where it is known that testing for convexity is NP-hard already in the case of quartic polynomials.