Log-concave polynomials, I: Entropy and a deterministic approximation algorithm for counting bases of matroids

We give a deterministic polynomial-time 2O(r)-approximation algorithm for the number of bases given matroid rank r and common any two matroids r. To best our knowledge, this is first nontrivial approximation that works arbitrary matroids. Based on lower bound Azar, Broder, Frieze, almost possible result assuming oracle access to independent sets matroid. There are main ingredients in result. For first, we build upon recent results Adiprasito, Huh, Katz, Wang combinatorial Hodge theory show basis generating polynomial (completely) log-concave polynomial. Formally, prove multivariate (and all its directional derivatives along positive orthant are) as functions over orthant. second ingredient, develop general framework approximate counting discrete problems, based convex optimization. The connection goes through subadditivity entropy. matroids, an superadditivity entropy holds by relying log-concavity