Improved bounds for some $S$-unit equations
Volume 214 / 2024
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 $.
