Improved bounds for some $S$-unit equations

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 $.