Integral points on generic fibers

Autores: Arnaud Bodin
Localización: Journal of the London Mathematical Society, ISSN 0024-6107, Vol. 81, Nº 3, 2010, págs. 563-572
Idioma: inglés
DOI: 10.1112/jlms/jdp084
Enlaces
- Texto completo (pdf)
Resumen
- Let P(x, y) be a rational polynomial. If the curve (P(x, y) = k), with k �¸ Q, is irreducible and admits an infinite number of points whose coordinates are integers, then Siegel�fs theorem implies that the curve is rational. We deal with the case where k is a generic value and prove, in the spirit of the Abhyankar.Moh.Suzuki theorem, that there exists an algebraic automorphism sending P(x, y) to the polynomial x or to x2 . y2, with �¸ N. Moreover for such curves we give a sharp bound for the number of integral points (x, y) with x and y bounded.