论文标题
Polya问题的足够条件
Sufficient conditions for a problem of Polya
论文作者
论文摘要
令$α$为非零代数数。让$ k $是$ \ mathbb {q}（α）$的GALOIS关闭，然后使用Galois Group $ G $，$ \ bar {\ Mathbb {Q}} $是$ \ Mathbb {q} $的代数关闭。在本文中，除其他结果外，我们证明了以下内容。 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代数整数。该结果概括了Polya [11]，Corvaja和Zannier [2]以及Philippon和Rath [9]的结果。我们还证明了该结果的类似于代数系数的合理函数。受B. de Smit [4]的结果的启发，我们证明了Polya型结果的有限版本，用于非零代数数的二进制复发序列。为了证明这些结果，我们应用了Corvaja和Zannier的技术以及Kulkarni等人[6]的结果，这是Schmidt子空间定理的应用。
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.
