Bounds for the solutions of superelliptic equations. (English) Zbl 0886.11016

Bugeaud, Yann

Compos. Math. 107, No. 2, 187-219 (1997).

Let \(f\) be a polynomial with integer coefficients in a number field \(K\), and consider the equation \(f(X)=Y^m\), where the variables \(X,Y\) are integers of \(K\) and \(m\geq 2\) is a rational integer. This equation was studied by several authors, the first effective bounds being given by A. Baker. The author improves the bounds of P. Voutier [J. Number Theory 53, 247-271 (1995; Zbl 0842.11008)] on the solutions of the equation, assuming that the polynomial satisfies LeVeque’s condition. Moreover, the arguments are extended to cover the S-integral solutions, as well. The main tool is the application of the results of Y. Bugeaud and K. Győry [Acta Arith. 74, 67-80 (1996; Zbl 0861.11023)] and [Acta Arith. 74, 273-292 (1996; Zbl 0861.11024)].

Reviewer: I.Gaál (Debrecen)

MSC:

11D61	Exponential Diophantine equations
11D75	Diophantine inequalities

Keywords:

superelliptic equation; Baker’s method

Citations:

Zbl 0842.11008; Zbl 0861.11023; Zbl 0861.11024