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Improved bounds for some $S$-unit equations

Volume 214 / 2024

Kálmán Győry, Samuel Le Fourn Acta Arithmetica 214 (2024), 311-326 MSC: Primary 11D61; Secondary 11D57, 11D59, 11J86 DOI: 10.4064/aa230530-24-8 Published online: 1 February 2024

Abstract

The $S$-unit equation $\alpha x + \beta y = 1$ in $x,y \in \mathcal O_S^\times $ plays a very important role in Diophantine number theory. We first present the best known effective upper bounds for the solutions of this equation, obtained recently by Le Fourn (2020) and Győry (2019). Then we prove some generalisations for the case of larger multiplicative groups instead of $\mathcal O_S^\times $. Further, we provide a new application to monic polynomials with given discriminant. Finally, we considerably improve our general upper bounds in the case of the special $S$-unit equation $x^n + y = 1$ in $x , y \in \mathcal O_S^\times $.

Authors

  • Kálmán GyőryUniversity of Debrecen
    H-4002 Debrecen, Hungary
    e-mail
  • Samuel Le FournUniv. Grenoble Alpes, CNRS, IF
    38000 Grenoble, France
    e-mail

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