Relations de Hodge--Riemann et combinatoire des matro\"ides (d'apr\`es K. Adiprasito, J. Huh et E. Katz)

TL;DR

This paper discusses the proof of Rota's conjecture on the log-concavity of matroid characteristic polynomial coefficients, using combinatorial algebraic geometry techniques inspired by algebraic geometry.

Contribution

It introduces a combinatorial Chow ring for matroids and proves analogs of key geometric theorems, leading to the proof of Rota's conjecture.

Findings

Proof of Rota's conjecture on log-concavity

Introduction of a combinatorial Chow ring for matroids

Establishment of Poincaré duality, Hard Lefschetz, and Hodge--Riemann relations for matroids

Abstract

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 Poincar\'e duality, the Hard Lefschetz theorem, and the Hodge--Riemann relations. The sought for log-concavity inequalities are then analogous to the Khovanskii--Teissier inequalities.