Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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

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.
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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