Recently, it was proved by Anari–Oveis Gharan–Vinzant, Anari–Liu–Oveis Gharan–Vinzant and Brändén–Huh that, for any matroid M, its basis generating polynomial and its independent set generating polynomial are log-concave on the positive orthant. Using these, they obtain some combinatorial inequalities on matroids including a solution of strong Mason's conjecture. In this paper, we study the strictness of the log-concavity of these polynomials and determine when equality holds in these combinatorial inequalities. We also consider a generalization of our result to morphisms of matroids.

中文翻译：

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拟阵的多项式的对数凹度的严格性

最近，由Anari–Oveis Gharan–Vinzant，Anari–Liu–Oveis Gharan–Vinzant和Brändén–Huh证明，对于任何拟阵M而言，其生成基多项式和其独立集生成多项式都是对数正凹的。使用这些，他们获得了拟阵上的一些组合不等式，包括强梅森猜想的解。在本文中，我们研究了这些多项式的对数凹度的严格性，并确定了何时在这些组合不等式中成立等式。我们还考虑将结果推广到类人动物的态射。