A note on Diophantine inequalities in function fields

Kathryn Wilson
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 30, 2024, Number 4, Pages 776–786
DOI: 10.7546/nntdm.2024.30.4.776-786
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Authors and affiliations

Kathryn Wilson
Department of Mathematics, Kansas State University
138 Cardwell Hall, 1228 N. 17th St, Manhattan, KS 66506, USA

Abstract

We will discuss how the Bentkus–Götze–Freeman variant of the Davenport–Heilbronn circle method can be used to study \mathbb{F}_q[t] solutions to inequalities of the form

    \[\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k-\gamma)<\tau , \]

where constants \lambda_1,\dots, \lambda_s \in\mathbb{F}_q((1/t)) satisfy certain conditions. This result is a generalization of the work done by Spencer in [11] to count the number of solutions to inequalities of the form

    \[\mathrm{ord}(\lambda_1p_1^k+\cdots+\lambda_sp_s^k)<\tau.\]

Keywords

  • Diophantine inequalities
  • Davenport–Heilbronn method
  • Hardy–Littlewood circle method
  • Function fields

2020 Mathematics Subject Classification

  • 11D75
  • 11P55
  • 11T55

References

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  2. Chevalley, C. (1935). Démonstration d’une hypothèse de M. Artin. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 11(1), 73–75.
  3. Davenport, H., & Heilbronn, H. (1946). On indefinite quadratic forms in five variables. Journal of the London Mathematical Society, 21, 185–193.
  4. Freeman, D. E. (2000). Asymptotic lower bounds for Diophantine inequalities. Mathematika, 47(1–2), 127–159.
  5. Freeman, D. E. (2002). Asymptotic lower bounds and formulas for Diophantine inequalities. In: Berndt, B. (Ed.). Number Theory for the Millennium, II, A K Peters, Natick, MA, 57–74.
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  7. Hsu, C.-N. (2001). Diophantine inequalities for the non-Archimedean line Fq((1/T )). Acta Arithmetica, 97(3), 253–267.
  8. Kubota, R. M. (1974). Waring’s problem for Fq[x]. Dissertationes Mathematicae (Rozprawy Matematyczne), 117, 60pp.
  9. Lê, T. H., Liu, Y.-R., & Wooley, T. D. (2023). Equidistribution of polynomial sequences in function fields, with applications. arXiv, Available online at: https://arxiv.org/abs/1311.0892
  10. Liu, Y.-R., & Wooley, T. D. (2010). Waring’s problem in function fields. Journal fur die reine und angewandte Mathematik, 638, 1–67.
  11. Spencer, C. V. (2009). Diophantine inequalities in function fields. Bulletin of the London Mathematical Society, 41(2), 341–353.
  12. Weil, A. (1949). Numbers of solutions of equations in finite fields. Bulletin of the American Mathematical Society, 55, 497–508.
  13. Wooley, T. D. (2003). On Diophantine inequalities: Freeman’s asymptotic formulae. Bonner Mathematischen Schriften, 360, Article 30.

Manuscript history

  • Received: 7 February 2024
  • Revised: 13 November 2024
  • Accepted: 14 November 2024
  • Online First: 18 November 2024

Copyright information

Ⓒ 2024 by the Author.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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Cite this paper

Wilson, K. (2024). A note on Diophantine inequalities in function fields. Notes on Number Theory and Discrete Mathematics, 30(4), 776-786, DOI: 10.7546/nntdm.2024.30.4.776-786.

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