Special 5-term recurrence relations, Banded Toeplitz matrices, and Reality of Zeros

TL;DR

This paper establishes conditions under which polynomials satisfying a five-term recurrence relation have all real zeros, linking their properties to the geometry of associated banded Toeplitz matrices and Laurent polynomial critical points.

Contribution

It provides new criteria for the reality of zeros of polynomials from five-term recurrences, connecting polynomial roots with the topology of level curves of related Laurent polynomials.

Findings

Zeros are real when critical points of $b(z)$ are all real.

Presence of specific Jordan curves in $b^{-1}( eals)$ ensures hyperbolicity.

Conditions on parameters determine the geometric structure of level curves.

Abstract

Below we establish the conditions guaranteeing the reality of all the zeros of polynomials $P_n(z)$ in the polynomial sequence $\{P_n(z)\}_{n=1}^{\infty}$ satisfying a five-term recurrence relation $$P_{n}(z)= zP_{n-1}(z) + \alpha P_{n-2}(z)+\beta P_{n-3}(z)+\gamma P_{n-4}(z),$$ with the standard initial conditions $$P_0(z) = 1, P_{-1}(z) = P_{-2}(z) =P_{-3}(z) = 0,$$ where $\alpha, \beta, \gamma$ are real coefficients, $\gamma\neq 0$ and $z$ is a complex variable. We interprete this sequence of polynomials as principal minors of an appropriate banded Teoplitz matrix whose associated Laurent polynomial $b(z)$ is holomorphic in $\mathbb{C}\setminus \{0\}$. We show that when either the critical points of $b(z)$ are all real; or when they are two real and one pair of complex conjugate critical points with some extra conditions on the parameters, the set $b^{-1}(\mathbb{R})$ contains a Jordan curve with $0$ in its interior and in some cases a non-simple curve enclosing $0$. The presence of the said curves is necessary and sufficient for every polynomial in the sequence $\{P_n(z)\}_{n=1}^{\infty}$ to be hyperbolic (real-rooted).