Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

2026/07/26 by Yanjun Han, Jonathan Niles-Weed
#math.CO #cs.IT #math.IT #math.PR

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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 χ² 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.

Citations

• Approximate independence of permutation mixtures
• Relative entropy bounds for sampling with and without replacement
• A Maclaurin type inequality
• An Information-Theoretic Proof of a Finite de Finetti Theorem
• Pseudorandom Generators for Width-3 Branching Programs
• Inequalities and tail bounds for elementary symmetric polynomial with applications
• On the Nuclear Norm and the Singular Value Decomposition of Tensors
• On a Remez-type inequality for trigonometric polynomials
• On Bobkov's approximate de Finetti representation via approximation of permanents of complex rectangular matrices

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• Power mean inequalities and sums of squares
• Symmetric polynomials and l^p inequalities for certain intervals of p
• The 'Core' of Symmetric Homogeneous Polynomial Inequalities of Degree Four of Three Real Variables