A Generalized Muirhead Inequality and Symmetric Sums of Nonnegative Circuits

TL;DR

This paper introduces a generalized Muirhead inequality to certify nonnegativity of symmetric polynomials and characterizes their decomposition into sums of nonnegative circuit polynomials, simplifying previous results.

Contribution

It extends classical inequalities to symmetric polynomials and provides a new, elementary proof for their nonnegativity certification via circuit polynomials.

Findings

Generalized Muirhead inequality for symmetric polynomials

Characterization of nonnegative symmetric polynomials via circuit decompositions

Simplified proof of existing nonnegativity results

Abstract

Circuit polynomials are a certificate of nonnegativity for real polynomials, which can be derived via a generalization of the classical inequality of arithmetic and geometric means. In this article, we show that similarly nonnegativity of symmetric real polynomials can be certified via a generalization of the classical Muirhead inequality. Moreover, we show that a nonnegative symmetric polynomial admits a decomposition into sums of nonnegative circuit polynomials if and only if it satisfies said generalized Muirhead condition. The latter re-proves a result by Moustrou, Naumann, Riener, Theobald, and Verdure for the case of the symmetric group in a shortened and more elementary way.