Bounds for the solutions of some diophantine equations in terms of discriminants

Abstract

Several effective upper bounds are known for the solutions of Thue equations, Thue-Mahler equations and superelliptic equations. One of the basic parameters occurring in these bounds is the height of the polynomial involved in the equation. In the present paper it is shown that better (and, in certain important particular cases, best possible) upper bounds can be obtained in terms of the height, if one takes into consideration also the discriminant of the polynomial.

On the magnitude of integer points on elliptic curves

“…However, Lang's conjecture seems well beyond the reach of current techniques. The purpose of this note is to point out that a recent result of [2] on Thue equations gives a reasonable bound which is polynomial in the height of / . By the Nagell-Lutz theorem, Lang's conjecture is true for trivial elliptic curves (when the Mordell-Weil group of the curve is trivial).…”

“…Let a and 6 be fixed in (1) and suppose that there exists a solution (xo,yo) to equation (1) such that J/Q does not divide the discriminant of / (for example, a = -1,6 = 1 and (xo,yo) = (56,419)). By using the Nagell-Lutz theorem again it is easy to see that the curves (2) y 2 = x 3 + an 2 x + bn 3 = /"(*), with (n,i/o) = 1 are non-trivial. Supposing n satisfies the inequality…”

Thue equations and CM-fields

“…Nevertheless, computation techniques for the resolution of Thue equations have been developed based on the above results [1], [13], [22], [28] and the solutions of certain parameterized families of Thue equations have been obtained [14]. Furthermore, upper bounds for the number of integral solutions of Thue equations have been given [5], [9], [4].…”

Thue equations and CM-fields

“…Nevertheless, computation techniques for the resolution of Thue equations have been developed based on the above results [1], [13], [22], [28] and the solutions of certain parameterized families of Thue equations have been obtained [14]. Furthermore, upper bounds for the number of integral solutions of Thue equations have been given [5], [9], [4].…”