Hodge-Riemann关系与矩阵的组合（d 'après K. Adiprasito，J. Huh et E.卡茨）
Relations de Hodge--Riemann et combinatoire des matroïdes (d'après K. Adiprasito, J. Huh et E. Katz)
摘要：
有限拟阵是表达线性独立性概念的组合结构。1964年，G. C. Rota证明了拟阵$M$的“特征多项式”的系数，多项式的系数枚举其给定秩的子集，形成对数凹序列。K. Adiprasito，J. Huh et E.卡茨证明了这一猜想使用的方法，虽然完全组合，启发代数几何。从伯格曼风扇的拟阵$M$，他们定义了一个分级的“周环”$A(M)$，他们证明类似的庞加莱对偶，硬莱夫谢茨定理，和霍奇-黎曼关系。求对数不等式则类似于Khovanskii-Teissier不等式。
英文摘要：
Finite matroids are combinatorial structures that express the concept of linear independence. In 1964, G.-C. Rota conjectured that the coefficients of the "characteristic polynomial" of a matroid $M$, polynomial whose coefficients enumerate its subsets of given rank, form a log-concave sequence. K. Adiprasito, J. Huh et E. Katz have proved this conjecture using methods which, although entirely combinatorial, are inspired by algebraic geometry. From the Bergman fan of the matroid $M$, they define a graded "Chow ring" $A(M)$ for which they prove analogs of the Poincaré duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.
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