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On a simple quartic family of Thue equations over imaginary quadratic number fields

Volume 208 / 2023

Benjamin Earp-Lynch, Bernadette Faye, Eva G. Goedhart, Ingrid Vukusic, Daniel P. Wisniewski Acta Arithmetica 208 (2023), 355-389 MSC: Primary 11D59; Secondary 11R11, 11Y50. DOI: 10.4064/aa230329-19-6 Published online: 8 September 2023

Abstract

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\epsilon }$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.

Authors

  • Benjamin Earp-LynchCarleton University
    Ottawa, ON, Canada
    e-mail
  • Bernadette FayeUFR SATIC
    Université Alioune Diop de Bambey
    Bambey 30, Diourbel, Sénégal
    e-mail
  • Eva G. GoedhartFranklin & Marshall College
    Lancaster, PA 17603, USA
    e-mail
  • Ingrid VukusicUniversity of Salzburg
    5020 Salzburg, Austria
    e-mail
  • Daniel P. WisniewskiDepartment of Mathematics/Computer Science
    DeSales University
    Center Valley, PA 18034, USA
    e-mail

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