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December 2008 Polynomial parametrization of the solutions of diophantine equations of genus 0
Sophie Frisch, Günter Lettl
Funct. Approx. Comment. Math. 39(2): 205-209 (December 2008). DOI: 10.7169/facm/1229696571


Let $f \in \mathbb{Z}[X,Y,Z]$ be a non-constant, absolutely irreducible, homogeneous polynomial with integer coefficients, such that the projective curve given by $f=0$ has a~function field isomorphic to the rational function field $\mathbb{Q} (T)$. We show that all integral solutions of the Diophantine equation $f=0$ (up to those corresponding to some singular points) can be parametrized by a single triple of integer-valued polynomials. In general, it is not possible to parametrize this set of solutions by a~single triple of polynomials with integer coefficients.


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Sophie Frisch. Günter Lettl. "Polynomial parametrization of the solutions of diophantine equations of genus 0." Funct. Approx. Comment. Math. 39 (2) 205 - 209, December 2008.


Published: December 2008
First available in Project Euclid: 19 December 2008

zbMATH: 1214.11043
MathSciNet: MR2490736
Digital Object Identifier: 10.7169/facm/1229696571

Primary: 11D85
Secondary: 11D41 , 13F20 , 14H05

Keywords: Diophantine equation , integer-valued polynomial , polynomial parametrization , resultant

Rights: Copyright © 2008 Adam Mickiewicz University

Vol.39 • No. 2 • December 2008
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