AI 中文总结
研究和为零的向量及行列和为零的方阵的初等对称多项式不等式，应用这些结果得到排列混合平均场近似保证的统一上界及德·菲内蒂定理的精确卡方版本，证明思路由GPT - 5.5 Pro模型提出。
AI 中文摘要
我们证明了对于和为零的向量以及行和与列和为零的方阵的初等对称多项式（ESPs）的新不等式。我们应用这些结果得到排列混合的平均场近似保证的统一上界，以及关于小字母表上有限序列的德·菲内蒂定理的一个精确的卡方版本。主要证明思路由GPT - 5.5 Pro模型提出。
英文摘要
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.
