对数凹性与全幺模多胞体的近似计数
Log-concavity and Approximate Counting for Totally Unimodular Polytopes
AI总结：
本文提出全幺模多胞体格点数的下界，基于Gurvits容量优化，实现近似计数算法，并证明相关猜想。
AI中文摘要：
我们提出了一个关于所有全幺模多胞体格点数的新的下界，推广了先前关于列联表、整数流及其他方面的下界。我们的下界基于Gurvits容量凸优化问题，因此我们的结果意味着一个有效的确定性算法，用于在显式指数因子内近似计数格点。我们通过证明相关的生成多项式属于一类新的对数凹多项式（称为VLC，即“变量式对数凹性”）来实现我们的下界。这也证明了Ferroni和Higashitani关于幺模多胞体Ehrhart多项式取值的猜想。这些结果的关键成分是解决了Barvinok关于直线上列联表的对数凹性猜想，该猜想是使用ChatGPT 6 Astra证明的。我们推测了Barvinok猜想的一个推广，我们认为这将导致更强、更一般的下界。
英文摘要：
We present a new lower bound on the number of lattice points of all totally unimodular polytopes, generalizing previous lower bounds on contingency tables, integer flows, and beyond. Our bound is based on the Gurvits capacity convex optimization problem, and thus our result implies an efficient deterministic algorithm for approximate counting of the lattice points up to an explicit exponential factor. We achieve our bounds by showing that the associated generating polynomials fit into a new general class of log-concave polynomials called VLC ("variable-wise log-concavity''). This also implies a conjecture of Ferroni and Higashitani on the evaluations of the Ehrhart polynomials of unimodular polytopes. The essential ingredient for these results is the resolution of Barvinok's log-concavity conjecture for contingency tables on lines, which was proven using ChatGPT 6 Astra. We conjecture a generalization of Barvinok's conjecture, which we believe will lead to stronger and more general bounds.
