Functiones et Approximatio Commentarii Mathematici

Polynomial parametrization of the solutions of diophantine equations of genus 0

Abstract

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.

Article information

Source
Funct. Approx. Comment. Math. Volume 39, Number 2 (2008), 205-209.

Dates
First available in Project Euclid: 19 December 2008

https://projecteuclid.org/euclid.facm/1229696571

Digital Object Identifier
doi:10.7169/facm/1229696571

Mathematical Reviews number (MathSciNet)
MR2490736

Zentralblatt MATH identifier
1214.11043

Citation

Frisch, Sophie; Lettl, Günter. Polynomial parametrization of the solutions of diophantine equations of genus 0. Funct. Approx. Comment. Math. 39 (2008), no. 2, 205--209. doi:10.7169/facm/1229696571. https://projecteuclid.org/euclid.facm/1229696571

References

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