Asymptotic equidistribution for partition statistics and topological invariants

Abstract

We provide a general framework for proving asymptotic equidistribution, convexity, and logconcavity of coefficients of generating functions on arithmetic progressions. Our central tool is a variant of Wright's Circle Method proved by two of the authors with Bringmann and Ono, following work of Ngo and Rhoades. We offer a selection of different examples of such results, proving asymptotic equidistribution results for several partition statistics, modular sums of Betti numbers of two-and three-flag Hilbert schemes, and the number of cells of dimension a (mod b) of a certain scheme central in work of Göttsche.

Jensen polynomials associated with Wright's circle method: Hyperbolicity and Turán inequalities

“…for every n ≥ 1. The second order Turán inequality is also commonly referred to as log-concavity and has been well studied in many settings [3], [5], [9], [10], [11]. For higher orders, a classical theorem of Hermite states that a polynomial J(X) with real coefficients is hyperbolic precisely when the associated Hankel matrix is positive-definite.…”