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Stern polynomials and algebraic independence

Daniel Duverney, Iekata Shiokawa

Abstract

Let $t\geq2$ and $k\geq1$ be integers. Let $H_{k}(z)$ with $\left\vert z\right\vert <1$ 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 $H_{k}(z^{t^{k}})$.

Stern polynomials and algebraic independence

Abstract

Let and be integers. Let with 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 and .

Paper Structure

This paper contains 2 sections, 6 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['ThStern']}

Key Result

Theorem 1.1

For any algebraic number $\alpha$ with $0<\left\vert \alpha\right\vert <1,$ the numbers $H_{k}(\alpha)$ and $H_{k}(\alpha^{t^{k}})$ are algebraically independent.

Theorems & Definitions (11)

  • Theorem 1.1
  • Corollary 1.1
  • Example 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 1 more