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arXiv · 2211.07266

A Generalized Muirhead Inequality and Symmetric Sums of Nonnegative Circuits

Janin Heuer·Ngoc Mai Tran·Timo de Wolff
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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.

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Janin Heuer, Ngoc Mai Tran, Timo de Wolff. 2022-11-14. A Generalized Muirhead Inequality and Symmetric Sums of Nonnegative Circuits. https://arxiv.org/abs/2211.07266

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