Analytic properties of Speyer's $g$-polynomial of uniform matroids

TL;DR

This paper investigates the properties of Speyer's $g$-polynomials for uniform matroids, proving they have only real zeros and that their coefficients are asymptotically normal, advancing understanding of their algebraic and probabilistic characteristics.

Contribution

It establishes recurrence relations for Speyer's $g$-polynomials and proves their real-rootedness and asymptotic normality of coefficients, which were previously unknown.

Findings

$g_{U_{n,d}}(t)$ has only real zeros for all valid $n,d$

Coefficients of $g_{U_{n,[n/2]}}(t)$ are asymptotically normal

Recurrence relations for Speyer's $g$-polynomials are derived

Abstract

Let $U_{n,d}$ denote the uniform matroid of rank $d$ on $n$ elements. We obtain some recurrence relations satisfied by Speyer's $g$-polynomials $g_{U_{n,d}}(t)$ of $U_{n,d}$. Based on these recurrence relations, we prove that the polynomial $g_{U_{n,d}}(t)$ has only real zeros for any $n-1\geq d\geq 1$. Furthermore, we show that the coefficient of $g_{U_{n,[n/2]}}(t)$ is asymptotically normal by local and central limit theorems.