Complexity of Log-concave Inequalities in Matroids

A sequence of nonnegative real numbers a1,a2,…,an, is log-concave if a2i≥ai−1ai+1 for all i ranging from 2 to n−1. Examples of log-concave inequalities range from inequalities that are readily provable, such as the binomial coefficients ai=(ni), to intricate inequalities that have taken decades to resolve, such as the number of independent sets ai in a matroid M with i elements (otherwise known as the first Mason's conjecture; and was resolved by June Huh in 2010s in a remarkable breakthrough). It is then natural to ask if it can be shown that the latter type of inequalities is intrinsically more challenging than the former. In this talk, we provide a rigorous framework to answer this type of questions, by employing a combination of combinatorics, complexity theory, and geometry.

This is a joint work with Igor Pak.