Zeros of Stern polynomials in the complex plane
David Altizio
TL;DR
This work investigates zeros of Stern polynomials $S_n(\lambda)$ defined by $S_0(\lambda)=0$, $S_1(\lambda)=1$, $S_{2n}(\lambda)=\lambda S_n(\lambda)$, and $S_{2n+1}(\lambda)=S_n(\lambda)+S_{n+1}(\lambda)$. It proves that no zeros lie in the disk $|w-2|\le 1$ by employing the Parabola Theorem for complex continued fractions to bound the corresponding continued-fraction expressions; this yields a significant restriction on zeros and advances the conjecture that all zeros lie in $\{\Re w<1\}$. A key methodological innovation is expressing ratios of Stern polynomials as continued fractions and using value/element sets $V_\alpha$ and $E_\alpha$ (with $\alpha=\pi/12$) to certify nonvanishing. As a corollary, the paper proves that $S_p(\lambda)$ is irreducible in $\mathbb{Z}[\lambda]$ for every prime $p$, resolving a conjecture of Ulas and Ulas with analytic continuation arguments rather than purely algebraic ones. These results sharpen our understanding of Stern polynomials' zeros and illustrate the power of complex-analytic continued-fraction techniques in algebraic questions.
Abstract
The classical Stern sequence of positive integers was extended to a polynomial sequence $S_n(λ)$ by Klavžar et. al. by defining $S_0(λ) = 0$, $S_1(λ) = 1$, and $$S_{2n}(λ) = λS_n(λ),\quad S_{2n+1}(λ) = S_n(λ) + S_{n+1}(λ).$$ Dilcher et. al. conjectured that all roots of $S_n(λ)$ 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 $S_n(λ)$. 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 $S_p(λ)$ is irreducible in $\mathbb Z[λ]$ whenever $p$ is a positive prime.