Log-concave polynomials III: Mason’s ultra-log-concavity conjecture for independent sets of matroids

IF 0.8 3区数学 Q2 MATHEMATICS

Proceedings of the American Mathematical Society Pub Date : 2024-03-20 DOI:10.1090/proc/16724

Nima Anari, Kuikui Liu, Shayan Oveis Gharan, Cynthia Vinzant

引用次数: 0

Abstract

We give a self-contained proof of the strongest version of Mason’s conjecture, namely that for any matroid the sequence of the number of independent sets of given sizes is ultra log-concave. To do this, we introduce a class of polynomials, called completely log-concave polynomials, whose bivariate restrictions have ultra log-concave coefficients. At the heart of our proof we show that for any matroid, the homogenization of the generating polynomial of its independent sets is completely log-concave.

查看原文本刊更多论文

对数凹多项式 III：梅森矩阵独立集的超对数凹猜想

我们给出了梅森猜想最强版本的自足证明，即对于任何矩阵，给定大小的独立集合数列是超对数凹的。为此，我们引入了一类多项式，称为完全对数凹多项式，它们的双变量限制具有超对数凹系数。我们证明的核心是，对于任何 matroid，其独立集的生成多项式的同调都是完全对数凹的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Proceedings of the American Mathematical Society 数学-数学

CiteScore

1.70

自引率

10.00%

发文量

207

审稿时长

2-4 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to shorter research articles (not to exceed 15 printed pages) in all areas of pure and applied mathematics. To be published in the Proceedings, a paper must be correct, new, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Longer papers may be submitted to the Transactions of the American Mathematical Society. Published pages are the same size as those generated in the style files provided for AMS-LaTeX.