Zeros of Hook Polynomials and Related Questions

TL;DR

This paper investigates the zeros of hook polynomials derived from partition statistics, providing asymptotic results and exploring related polynomial families, including deformations of Rogers-Ramanujan functions.

Contribution

It offers new asymptotic analysis of hook polynomials and addresses open questions about their zeros, extending previous research in the area.

Findings

Asymptotics for $t$-hook polynomial values outside an annulus

Identification of isolated zeros of a theta function

Data on polynomial families related to Rogers-Ramanujan deformations

Abstract

We study the zero set of polynomials built from partition statistics, complementing earlier work in this direction by Boyer, Goh, Parry, and others. In particular, addressing a question of Males with two of the authors, we prove asymptotics for the values of $t$-hook polynomials away from an annulus and isolated zeros of a theta function. We also discuss some open problems and present data on other polynomial families, including those associated to deformations of Rogers-Ramanujan functions.