Improved bounds for some \(S\)-unit equations (Q6595589)

Let \(K\) be an algebraic number field and \(S\) a finite set of places of \(K\) containing all infinite places. Denote by \( \mathcal{O}_S \) the ring of \(S\)-integers, and by \( \mathcal{O}^{\times}_S \) the group of \(S\)-units in \(K\). Let \( \alpha \) and \( \beta \) be nonzero elements of \(K\) and consider the \(S\)-unit equation \(\alpha x+ \beta y =1 ~ \text{in} ~ x,y \in \mathcal{O}^{\times}_S\), which plays a very important role in Diophantine number theory. \N\NIn the paper under review, the authors first present the best known effective upper bounds for the solutions of the \( S \)-unit equation, obtained recently by \textit{S. Le Fourn} [Algebra Number Theory 14, No. 3, 785--807 (2020; Zbl 1446.11125)] and \textit{K. Győry} [Publ. Math. Debr. 94, No. 3--4, 507--526 (2019; Zbl 1438.11084)]. Then they prove some generalisations for the case of larger multiplicative groups instead of \( \mathcal{O}^{\times}_S \). Furthermore, they provide a new application to monic polynomials with given discriminant. Finally, they considerably improve their general upper bounds in the case of the special \(S\)-unit equation \(x^n + y = 1\) in \(x,y\in \mathcal{O}^{\times}_S \).