Diophantine equations in separated variables
Let \(K\) be a number field, a finite set \(S\) of places of \(K\) that contains all Archimedean places and the ring \(O_S\) of \(S\)-integers of \(K\), and \(f, g\in K[x]\). The study of solutions in \(O_S\) of Diophantine equations of type \(f(x)= g(y)\) have been of long-standing interest to number theorists and several important results have been obtained \textit{D. Kreso} and \textit{R. F. Tichy} [Grazer Math. Ber. 363, 143--170 (2015; Zbl 1395.11054)]. In this paper, some standard methods of Galois theory for maps between curves are used to give simple and unifying proof of most of these results. For a polynomial \(f\), the roots of the derivative \(f^{\prime}\) are called critical points, and the values of \(f\) at critical points are called critical values. If for critical points \(b_i\) 's of \(f\), one has \(f(b_i) \neq f(b_j)\) when \(b_i \neq b_j\), then \(f\) is said to have all distinct critical values. In this paper it is proved that if \( \deg f \geq 3\), \(\deg g \geq 3\), \(f\) and \(g\) both have at least two distinct critical points and all distinct critical values, then the equation \(f(x) = g(y)\) has infinitely many solutions \(x\), \(y\) with a bounded \(O_S\)-denominator (i.e. there is a nonzero \(\delta\in O_S\) such \(\delta x, \delta y \in O_S\)) if and only if \(f(x) = g(\mu(x))\) for some linear \(\mu \in K[x]\). Furthermore, it is proved that if both \(f\) and \(g\) have at least two distinct critical points and equal critical values at at most two distinct critical points, and their derivatives do not satisfy some condition, then the equation \(f (x) = g(y)\) has only finitely many solutions with a bounded \(O_S\)-denominator, unless either \((\deg f, \deg g) = (3, 5)\), or \(f\) is indecomposable and \(g(x) = f (\nu(x))\) for some quadratic \(\nu\in K[x]\). Moreover, some interesting corollaries have been given.
scientific article; zbMATH DE number 6865418
scientific article