Functiones et Approximatio Commentarii Mathematici

Polynomial parametrization of the solutions of diophantine equations of genus 0

Sophie Frisch and Günter Lettl

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.facm/1229696571

Digital Object Identifier
doi:10.7169/facm/1229696571

Mathematical Reviews number (MathSciNet)
MR2490736

Zentralblatt MATH identifier
1214.11043

Subjects
Primary: 11D85: Representation problems [See also 11P55]
Secondary: 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25] 11D41: Higher degree equations; Fermat's equation 14H05: Algebraic functions; function fields [See also 11R58]

Keywords
Diophantine equation integer-valued polynomial resultant polynomial parametrization

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


Export citation

References

  • S. Frisch, Remarks on polynomial parametrization of sets of integer points, Comm. Algebra 36 (2008), 1110--1114.
  • S. Frisch, L. Vaserstein, Parametrization of Pythagorean triples by a single triple of polynomials, J. Pure Appl. Algebra 212 (2008), 271--274.
  • E. Kunz, Introduction to Plane Algebraic Curves, Birkhäuser, 2005.
  • D. Poulakis, E. Voskos, Solving genus zero Diophantine equations with at most two infinite valuations, J. Symbolic Computation 33 (2002), 479--491.
  • R.,J. Walker, Algebraic Curves, Springer, 1978.