Connections between discriminants and the root distribution of polynomials with rational generating function

TL;DR

This paper explores the relationship between discriminants and root distributions of certain polynomial sequences with rational generating functions, showing roots lie on explicit algebraic curves for specific cases.

Contribution

It establishes a connection between discriminants and root locations for polynomials with generating functions as reciprocals of specific bivariate polynomials, extending previous understanding.

Findings

Roots lie on explicit algebraic curves for given polynomial cases

The approach involves the q-analogue of the discriminant

Results apply to polynomials with generating functions as reciprocals of quadratic, cubic, and quartic polynomials

Abstract

Let $H_{m}(z)$ be a sequence of polynomials whose generating function $\sum_{m=0}^{\infty}H_{m}(z)t^{m}$ is the reciprocal of a bivariate polynomial $D(t,z)$. We show that in the three cases $D(t,z)=1+B(z)t+A(z)t^{2}$, $D(t,z)=1+B(z)t+A(z)t^{3}$ and $D(t,z)=1+B(z)t+A(z)t^{4}$, where $A(z)$ and $B(z)$ are any polynomials in $z$ with complex coefficients, the roots of $H_{m}(z)$ lie on a portion of a real algebraic curve whose equation is explicitly given. The proofs involve the $q$-analogue of the discriminant, a concept introduced by Mourad Ismail.