Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

Yanjun Han, Jonathan Niles-Weed

arXiv:2607.23836v1 Primary: math.CO Secondary: cs.IT, math.PR

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.

Disclosure

“ain 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.”
arXiv metadata: abstract

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:07:47.299346Z

Structural counts

Pages 14 pdf

Theorems 2 source

Lemmas 1 source

Propositions 0 source

Corollaries 3 source

Definitions 0 source

Displayed equations 62 source

Bibliography entries 92 source

Appendix pages 3 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.