Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 026, 14 pages      arXiv:2408.16902      https://doi.org/10.3842/SIGMA.2025.026
Contribution to the Special Issue on Basic Hypergeometric Series Associated with Root Systems and Applications in honor of Stephen C. Milne

Zeros of Hook Polynomials and Related Questions

Walter Bridges a, William Craig b, Amanda Folsom c and Larry Rolen d
a) University of North Texas, USA
b) United States Naval Academy, USA
c) Amherst College, USA
d) Vanderbilt University, USA

Received September 03, 2024, in final form April 10, 2025; Published online April 21, 2025

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.

Key words: integer partitions; hook length; zeros of polynomials; zero attractor; asymptotic behavior; theta functions.

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