Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

Yanjun Han; Jonathan Niles-Weed

Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

Combinatorics 2026-07-26 v1 Information Theory Probability

Authors: Yanjun Han , Jonathan Niles-Weed

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 $\chi^2$ 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.

Cite

@article{arxiv.2607.23836,
  title  = {Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices},
  author = {Yanjun Han and Jonathan Niles-Weed},
  journal= {arXiv preprint arXiv:2607.23836},
  year   = {2026}
}