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Sufficient conditions for a problem of Polya

Abhishek Bharadwaj, Veekesh Kumar, Aprameyo Pal, R. Thangadurai

Abstract

Let $α$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb{Q}(α)$ with Galois group $G$ and $\bar{\mathbb{Q}}$ be the algebraic closure of $\mathbb{Q}$. In this article, among the other results, we prove the following. If $f\in \bar{\mathbb{Q}}[G]$ is a non-zero element of the group ring $\bar{\mathbb{Q}}[G]$ and $α$ is a given algebraic number such that $f(α^n)$ is a non-zero algebraic integer for infinitely many natural numbers $n$, then $α$ is an algebraic integer. This result generalizes the result of Polya [11], Corvaja and Zannier [2] and Philippon and Rath [9]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [4], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al., [6] which are applications of the Schmidt subspace theorem.

Sufficient conditions for a problem of Polya

Abstract

Let be a non-zero algebraic number. Let be the Galois closure of with Galois group and be the algebraic closure of . In this article, among the other results, we prove the following. If is a non-zero element of the group ring and is a given algebraic number such that is a non-zero algebraic integer for infinitely many natural numbers , then is an algebraic integer. This result generalizes the result of Polya [11], Corvaja and Zannier [2] and Philippon and Rath [9]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [4], we prove a finite version of the Polya type result for a binary recurrence sequences of non-zero algebraic numbers. In order to prove these results, we apply the techniques of Corvaja and Zannier along with the results of Kulkarni et al., [6] which are applications of the Schmidt subspace theorem.
Paper Structure (10 sections, 16 theorems, 54 equations)

This paper contains 10 sections, 16 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Our Results
  3. Polynomial Values :
  4. Group Rings
  5. Action under the trace map
  6. Determining the nature of rational functions
  7. Determining algebraic integers in finite iteration
  8. Preliminaries
  9. Proofs of Theorems 2.1 to \ref{['rationalfunction']}
  10. Proofs of Theorems \ref{['thm1.1']} and \ref{['finite']}

Key Result

Theorem 2.1

Let $\mathcal{L}(X_1,\dots,X_k) = \sum_{i=1}^k \lambda_i X_i$ be a linear form with coefficients $\lambda_i$ in $\overline{\mathbb{Q}}^\times$. Let $(\alpha_1, \ldots, \alpha_k) \in \bar{\mathbb{Q}}^{\times k}$ be a given $k$-tuple of algebraic numbers such that for $n$ in an infinite set $\mathfrak{S} \subset \mathbb{N}$. Then for each subset $I_j$ of $\mathcal{I} = \{1,\dots, k \}$ correspondin

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Remark 2.2
  • Theorem 2.4
  • Theorem 2.5
  • Corollary 2.1
  • Theorem 2.6
  • Theorem 2.7
  • ...and 12 more