Zeros of Stern polynomials in the complex plane

TL;DR

This paper investigates the roots of Stern polynomials in the complex plane, proving certain regions free of roots and confirming conjectures about their irreducibility for prime indices.

Contribution

It advances understanding of Stern polynomial roots by proving specific regions contain no roots and confirming irreducibility for prime-indexed polynomials.

Findings

No roots in the region {|w-2| ≤ 1} in the complex plane.

Confirmed the irreducibility of S_p(λ) for prime p.

Progress towards the conjecture that all roots lie in Re(w) < 1.

Abstract

The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(\lambda)$ by Klav\v{z}ar et. al. by defining $S_0(\lambda) = 0$, $S_1(\lambda) = 1$, and $$S_{2n}(\lambda) = \lambda S_n(\lambda),\quad S_{2n+1}(\lambda) = S_n(\lambda) + S_{n+1}(\lambda).$$ Dilcher et. al. conjectured that all roots of $S_n(\lambda)$ lie in the half-plane $\{\operatorname{Re} w < 1\}$. We make partial progress on this conjecture by proving that $\{|w-2| \leq 1\}\subseteq\mathbb C$ does not contain any roots of $S_n(\lambda)$. Our proof uses the Parabola Theorem for convergence of complex continued fractions. As a corollary, we establish a conjecture of Ulas and Ulas by showing that $S_p(\lambda)$ is irreducible in $\mathbb Z[\lambda]$ whenever $p$ is a positive prime.