On the location of ratios of zeros of special trinomials

TL;DR

This paper characterizes the zero distribution of special polynomial sequences defined by three-term recurrences and reveals geometric properties of zeros of related trinomials, including their location on algebraic curves and ratios on the real line or unit circle.

Contribution

It provides a full characterization of the zero locus for a class of polynomial sequences and uncovers new geometric properties of zeros of certain trinomials.

Findings

Zeros of the polynomial sequence lie on a specific real algebraic curve.

For any zero of the sequence, at least two zeros of an associated trinomial have ratios on the real line or unit circle.

The geometric properties of zeros are linked to the parameters of the recurrence and the form of the trinomial.

Abstract

Given coprime integers $k, \ell$ with $k > \ell \geqslant 1$ and arbitrary complex polynomials $A(z), B(z)$ with $\deg(A(z)B(z))\geqslant 1$, we consider the polynomial sequence $\{P_n(z)\}$ satisfying a three-term recurrence $P_n(z)+B(z)P_{n-\ell}(z)+A(z)P_{n-k}(z)=0$ subject to the initial conditions $P_0(z)=1$, $P_{-1}(z)=\cdots=P_{1-k}(z)=0$ and fully characterize the real algebraic curve $\Gamma$ on which the zeros of the polynomials in $\{P_n(z)\}$ lie. In addition, we show that, for any (randomly chosen) $n\in \mathbb{Z}_{\geqslant 1}$ and zero $z_0$ of $P_n(z)$ with $A(z_0)\neq 0$, at-least two of the distinct zeros of the trinomial $D(t;z_0):={A(z_0)t^{k}+ B(z_0)t^{\ell}+1} $ have a ratio that lies on the real line and / or on the unit circle centred at the origin. This reveals a previously unknown geometric property exhibited by the zeros of trinomials of the form $t^k+at^{\ell}+1$ where $a\in \mathbb{C}-\{0\}$ is such that $a^k\in \mathbb{R}$.