Sufficient conditions for a problem of Polya
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- by Abhishek Bharadwaj, Aprameyo Pal, Veekesh Kumar and R. Thangadurai;
- Proc. Amer. Math. Soc. 152 (2024), 3715-3730
- DOI: https://doi.org/10.1090/proc/16826
- Published electronically: July 19, 2024
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Abstract:
Let $\alpha$ be a non-zero algebraic number. Let $K$ be the Galois closure of $\mathbb {Q}(\alpha )$ 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 $\alpha$ is a given algebraic number such that $f(\alpha ^n)$ is a non-zero algebraic integer for infinitely many natural numbers $n$, then $\alpha$ is an algebraic integer. This result generalises the result of Polya [Rend. Circ Mat. Palermo, 40 (1915), pp. 1–16], Corvaja and Zannier [Acta Math. 193 (2004), pp. 175–191] and Philippon and Rath [J. Number Theory 219 (2021), pp. 198–211]. We also prove the analogue of this result for rational functions with algebraic coefficients. Inspired by a result of B. de Smit [J. Number Theory 45 (1993), pp. 112–116], 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. [Trans. Amer. Math. Soc. 371 (2019), pp. 3787–3804], which are applications of the Schmidt subspace theorem.References
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Bibliographic Information
- Abhishek Bharadwaj
- Affiliation: Department of Mathematics, Jeffery Hall, 99 University Avenue, Queen’s University, Kingston, Ontario K7L 3N6, Canada
- MR Author ID: 1211514
- Email: atb4@queensu.ca
- Aprameyo Pal
- Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj 211019, India
- MR Author ID: 1061635
- Email: aprameyopal@hri.res.in
- Veekesh Kumar
- Affiliation: Department of Mathematics, Indian Institute of Technology Dharwad, Chikkamalligawad village, Dharwad, Karnataka 580007, India
- MR Author ID: 1276667
- Email: veekeshk@iitdh.ac.in
- R. Thangadurai
- Affiliation: Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj 211019, India
- MR Author ID: 601482
- Email: thanga@hri.res.in
- Received by editor(s): February 15, 2023
- Received by editor(s) in revised form: September 13, 2023, February 20, 2024, and February 22, 2024
- Published electronically: July 19, 2024
- Communicated by: David Savitt
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 3715-3730
- MSC (2020): Primary 11J87; Secondary 11S99
- DOI: https://doi.org/10.1090/proc/16826
- MathSciNet review: 4781968