Rota's Conjecture and Positivity of Algebraic Cycles in Permutohedral Varieties.

Abstract

Rota's conjecture predicts that the coefficients of the characteristic polynomial of a matroid form a log-concave sequence. We give a proof of the conjecture for realizable matroids using techniques of algebraic geometry. The same approach to the conjecture in the general case (for possibly non-realizable matroids) leads to several intriguing questions on higher codimension algebraic cycles in the toric variety associated to the permutohedron.