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

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^∞ , whose terms are eventually hyperbolic. We also identify the real interval I ⊂ℝ^+ in which the collection of zeros ∪ _m ≫ 1𝒵(H_m(z)) forms a dense set.