On the number of \(S\)-integral solutions to \(Y^m = f(X)\) (Q1345722)

Let \(K\) be an algebraic number field, \(m\) an integer \(\geq 2\) and \(f(X)\) a non-zero polynomial with coefficients in \(K\). In 1986 \textit{J. H. Evertse} and \textit{J. Silverman} established the first explicit upper bound on the number of \(S\)-integral solutions to the equation \(Y^m= f(X)\), under some necessary and sufficient conditions for the curve described by this equation to be of positive genus [Math. Proc. Camb. Philos. Soc. 100, 237--248 (1986; Zbl 0611.10009)]. In the paper under review the author, using the method of Evertse and Silverman, gives some refinements and generalizations of their results. Also, a conjecture of Schmidt, in some particular cases, is proved.