Parametrized solutions of Diophantine equations. (English) Zbl 1108.11028

Lettl, Günter

Math. Slovaca 54, No. 5, 465-471 (2004).

Summary: In 1993, E. Thomas conjectured that for certain families of Thue equations there are – up to solutions arising from finitely many polynomials over the integers \(\mathbb Z\) – only finitely many further solutions over the integers. We give an example which shows that this conjecture cannot hold for arbitrary families of Thue equations where the coefficients are polynomials in one variable over the rational integers. Therefore we introduce the notion of \(\mathbb Z\)-parameter solutions of a family of Diophantine equations, which means a solution in algebraic functions which has infinitely many specializations to rational integers. With this revised setting, one might ask whether Thomas’s conjecture holds for families of Thue equations. Using C. L. Siegel’s theorem on integral points on an algebraic curve and an idea going back to E. Maillet, we prove some general results showing that \(\mathbb Z\)-parameter solutions generate very special function fields and have a very clear shape.

MSC:

11D59	Thue-Mahler equations
14G05	Rational points

Keywords:

Thue equation; Thomas’s conjecture; Siegel’s theorem; integral points

References:

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