On explicit bounds for the solutions of a class of parametrized Thue equations of arbitrary degree. (English) Zbl 1009.11023

Heuberger, Clemens

Monatsh. Math. 132, No. 4, 325-339 (2001).

Consider the equation \[ F_a(x,y)=\prod_{i=1}^n(x-p_i(a)y)-y^n=\pm 1 (x,y\in \mathbb Z) \] where \(a\) is an integer parameter, \(p_1,\ldots,p_n\) are monic polynomials. The author [Acta Arith. 98, 375-394 (2001; Zbl 0973.11043)] proved that (under some technical conditions) there exists a computable constant \(a_0\) depending on \(p_1,\ldots,p_n\) such that for \(a>a_0\) the above Thue equation has only solutions with \(|y|\leq 1\). In the present paper an explicit estimate is given for \(a_0\). It is important to stress that the result concerns a family of Thue equations of arbitrary high degrees: in this respect the article has a special importance.

Reviewer: Istvan Gaál (Debrecen)

MSC:

11D59	Thue-Mahler equations
11D41	Higher degree equations; Fermat’s equation

Keywords:

Thue equations; linear forms in the logarithms

Citations:

Zbl 0973.11043