Year: 2010
Type: article
Abstract: 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's 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 − ℓ y2, with ℓ ∈ ℕ. Moreover for such curves we give a sharp bound for the number of integral points (x, y) with x and y bounded.
DOI: 10.1112/jlms/jdp084
Author: Arnaud Bodin
Institutions: Université de Lille, Laboratoire Paul Painlevé, Université Lille 1
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2018233823
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