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Zeros of Hook Polynomials and Related Questions

Walter Bridges, William Craig, Amanda Folsom, Larry Rolen

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.

Zeros of Hook Polynomials and Related Questions

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 -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.
Paper Structure (11 sections, 6 theorems, 26 equations, 6 figures)

This paper contains 11 sections, 6 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Asymptotics for hook-length polynomials
  3. The case $\boldsymbol{|w|>1}$
  4. The case $\boldsymbol{|w|<1}$
  5. Other polynomials
  6. Polynomials from products
  7. Polynomials from Rogers--Ramanujan series
  8. Asymptotically real
  9. Imaginary
  10. Radial segments
  11. Circular

Key Result

Theorem 1.1

We have for $t \geq 2$ that

Figures (6)

  • Figure 1: Zeros of $P_7(w,n)$ for $n=425,985$ and $1405$.
  • Figure 2: Zeros of $P_7(w,n)$ for $n=426$, $986$ and $1406$.
  • Figure 3: Zeros of $p_{1,0}(w;n)$ with restricted real parts, for $n=1000, 5000$, and $10000$.
  • Figure 4: Zeros of $p_{1/2,1}(w;2000)$ with restricted imaginary parts.
  • Figure 5: Zeros of $p_{1,1/3}(w;2000)$ (left) and $p_{1,1/4}(w;2000)$ (right) with restricted moduli.
  • ...and 1 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Theorem 2.1: Laplace's method, see Pinsky
  • Proposition 2.2: Anderson
  • Claim 3.2