[Paper Review] Stern polynomials and algebraic independence

Daniel Duverney, Iekata Shiokawa|arXiv (Cornell University)|Mar 26, 2026

Advanced Combinatorial Mathematics0 citations

TL;DR

The authors prove that for integers t≥2, k≥1 and any nonzero algebraic α with |α|<1, the values H_k(α) and H_k(α^{t^k}) are algebraically independent, using Mahler’s method applied to associated continued fractions.

ABSTRACT

Let $t\geq2$ and $k\geq1$ be integers. Let $H_{k}(z)$ with $\left\vert z ight\vert &lt;1$ be the limit of a certain subsequence of the Stern polynomials introduced by Dilcher and Eriksen. We use Mahler's method to prove the algebraic independence of the values at nonzero algebraic points of the functions $H_{k}(z)$ and $H_{k}(z^{t^{k}})$.

Motivation & Objective

• Motivate the study of Stern polynomials and their limit functions H_k(z) within Diophantine and transcendence contexts.
• Aim to establish algebraic independence of the pair (H_k(α), H_k(α^{t^k})) for nonzero algebraic α with 0<|α|<1.
• Leverage continued fraction representations and Mahler’s method to derive independence results and transcendence consequences.

Proposed method

• Define H_k(z) as the limit of a subsequence of Dilcher–Eriksen Stern polynomials.
• Derive a functional system H_k(z^{t^k}), H_k(z^{t^{2k}}), etc., leading to a 2x2 matrix recurrence A(z).
• Use Mahler’s method for algebraic independence of functions satisfying a linear functional equation with an invertible specialization.
• Prove intermediate lemmas: A(z) has only 0 as a pole; H_k(1/2)/H_k(1/2^{t^k}) is irrational; H_k(z) is transcendental; H_k(z) and H_k(z^{t^k}) are algebraically independent over C(z).
• Apply [Ku. Nishioka, Mahler functions and transcendence] framework to conclude algebraic independence of the pair at algebraic α with 0<|α|<1.
• Derive corollaries about the transcendence of certain continued fractions involving H_k(α) / H_k(α^{t^k}).

Experimental results

Research questions

• RQ1 Do H_k(α) and H_k(α^{t^k}) form an algebraically independent pair for every nonzero algebraic α with 0<|α|<1?
• RQ2 Can Mahler’s method be effectively applied to the continued fraction representations arising from Stern polynomials to deduce transcendence results?
• RQ3 What are the transcendence consequences for the associated continued fractions and their specializations at algebraic points?
• RQ4 How do the properties of A(z) and the related matrix recurrence drive independence conclusions?

Key findings

• H_k(α) and H_k(α^{t^k}) are algebraically independent for every algebraic α with 0<|α|<1.
• The continued fraction expression for H_k(α)/H_k(α^{t^k}) converges and yields a transcendental value when α is algebraic with 0<|α|<1.
• H_k(z) is transcendental over C(z), and its coefficient structure is restricted to {0,1}.
• Az(z)–based functional system enables an application of Mahler’s method to deduce algebraic independence.
• Corollaries include transcendence results for the continued fraction representations involving H_k at algebraic points.