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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On covering systems of polynomial rings over finite fields
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by Huixi Li, Biao Wang, Chunlin Wang and Shaoyun Yi;
Proc. Amer. Math. Soc. 152 (2024), 3731-3742
DOI: https://doi.org/10.1090/proc/16864
Published electronically: July 1, 2024

Abstract:

In 1950, Erdős posed a question known as the minimum modulus problem on covering systems for $\mathbb {Z}$, which asked whether the minimum modulus of a covering system with distinct moduli is bounded. This long-standing problem was finally resolved by Hough [Ann. of Math. (2) 181 (2015), no. 1, pp. 361–382] in 2015, as he proved that the minimum modulus of any covering system with distinct moduli does not exceed $10^{16}$. Recently, Balister, Bollobás, Morris, Sahasrabudhe, and Tiba [Invent. Math. 228 (2022), pp. 377–414] developed a versatile method called the distortion method and significantly reduced Hough’s bound to $616,000$. In this paper, we apply this method to present a proof that the smallest degree of the moduli in any covering system for $\mathbb {F}_q[x]$ of multiplicity $s$ is bounded by a constant depending only on $s$ and $q$. Consequently, we successfully resolve the minimum modulus problem for $\mathbb {F}_q[x]$ and disprove a conjecture by Azlin [Covering Systems of Polynomial Rings Over Finite Fields, University of Mississippi, Electronic Theses and Dissertations. 39, 2011].
References
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Bibliographic Information
  • Huixi Li
  • Affiliation: School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, People’s Republic of China
  • ORCID: 0000-0003-4447-1157
  • Email: lihuixi@nankai.edu.cn
  • Biao Wang
  • Affiliation: School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, People’s Republic of China
  • MR Author ID: 1397071
  • ORCID: 0000-0003-1105-6299
  • Email: bwang@ynu.edu.cn
  • Chunlin Wang
  • Affiliation: School of Mathematical Sciences, Sichuan Normal University, Chengdu 610064, People’s Republic of China
  • MR Author ID: 1029427
  • Email: c-l.wang@outlook.com
  • Shaoyun Yi
  • Affiliation: School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China
  • MR Author ID: 1404024
  • ORCID: 0000-0002-1251-8611
  • Email: yishaoyun926@xmu.edu.cn
  • Received by editor(s): August 29, 2023
  • Received by editor(s) in revised form: November 26, 2023, and February 23, 2024
  • Published electronically: July 1, 2024
  • Additional Notes: The research of the first author was partially supported by the National Natural Science Foundation of China (Grant No. 12201313) and the Fundamental Research Funds for the Central Universities, Nankai University (Grant No. 63231145). The fourth author was supported by the National Natural Science Foundation of China (No. 12301016) and the Fundamental Research Funds for the Central Universities (No. 20720230025).
    The second author is the corresponding author.
  • Communicated by: Amanda Folsom
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3731-3742
  • MSC (2020): Primary 05B40, 11A07, 11B25, 11T06, 11T55
  • DOI: https://doi.org/10.1090/proc/16864
  • MathSciNet review: 4781969