Root-freeness and generalized log-concavity

Algebra Seminar
Monday, April 3, 2023 - 10:20am for 1.5 hours (actually 80 minutes)
Skiles 005
Thuy-Duong Vuong – Standford University – tdvuong@stanford.edu
Abeer Al Ahmadieh

In this talk, we explore the fractional log-concavity property of generating polynomials of discrete distributions. This property is an analog to the Lorentzian [Branden-Huh’19]/log-concavity [Anari-Liu-OveisGharan-Vinzant’19] property of the generating polynomials of matroids. We show that multivariate generating polynomials without roots in a sector of the complex plane are fractionally log-concave. Furthermore, we prove that the generating polynomials of linear delta matroids and of the intersection between a linear matroid and a partition matroid have no roots in a sector, and thus are fractionally log-concave. Beyond root-freeness, we conjecture that for any subset F of {0,1}^n such that conv(F) has constantly bounded edge length, the generating polynomial for the uniform distribution over F is fractionally log-concave.

Based on joint works with Yeganeh Alimohammadi , Nima Anari and Kirankumar Shiragur.