Sufficient conditions for a problem of Polya
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866917596185493504 |
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| author | Bharadwaj, Abhishek Kumar, Veekesh Pal, Aprameyo Thangadurai, R. |
| author_facet | Bharadwaj, Abhishek Kumar, Veekesh Pal, Aprameyo Thangadurai, R. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_04452 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sufficient conditions for a problem of Polya Bharadwaj, Abhishek Kumar, Veekesh Pal, Aprameyo Thangadurai, R. Number Theory Primary 11J87, Secondary 11S99 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. |
| title | Sufficient conditions for a problem of Polya |
| topic | Number Theory Primary 11J87, Secondary 11S99 |
| url | https://arxiv.org/abs/2202.04452 |