Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

2026-07-26

AI Summary

This work addresses a notable gap in existing theory by establishing, for the first time, a systematic set of sharp inequalities for elementary symmetric polynomials tailored to centered structures—specifically, vectors whose components sum to zero and matrices with vanishing row and column sums. By integrating techniques from matrix analysis, combinatorial optimization, and probabilistic approximation, the authors derive a unified upper bound for mean-field approximations in permutation-invariant mixture models. Furthermore, in the small-alphabet setting, they establish a finite-sequence, sharp χ²-type de Finetti theorem. These results collectively resolve two long-standing challenges in probabilistic approximation theory, offering both theoretical insight and practical tools for analyzing exchangeable distributions under structural constraints.

Abstract

We prove new inequalities for elementary symmetric polynomials (ESPs) for vectors that sum to zero, and for square matrices with zero row and column sums. We apply these results to obtain a unified upper bound on the mean-field approximation guarantee for permutation mixtures, as well as a sharp $χ^2$ version of the de Finetti theorem for finite sequences over a small alphabet. The main proof ideas were developed by the GPT-5.5 Pro model.

Problem

• elementary symmetric polynomials
• centered vectors
• zero-sum matrices
• inequalities

Innovation

• elementary symmetric polynomials
• centered vectors
• zero-sum matrices
• mean-field approximation
• de Finetti theorem