Zeros of Polynomials Generated by a Rational Function with a Hyperbolic-Type Denominator

Abstract

This paper investigates the zero locus of a sequence of polynomials generated by a bivariate rational function with a denominator of the form $$G(z,t)=P(t)+zt^{r}$$ , where the zeros of P are positive and real. We show that every member of a family of such generating functions—parametrized by the degree of P and r—gives rise to a sequence of polynomials $$\{H_{m}(z)\}_{m=0}^{\infty }$$ , whose terms are eventually hyperbolic. We also identify the real interval $$I \subset \mathbb {R}^+$$ in which the collection of zeros $$\cup _{m \gg 1} \mathcal {Z}(H_m(z))$$ forms a dense set.