Polynomial parametrization of the solutions of Diophantine equations of genus 0

TL;DR

This paper demonstrates that solutions to certain genus 0 Diophantine equations can be parametrized by a single triple of integer-valued polynomials, providing a constructive approach for solutions up to singular points.

Contribution

It introduces a method to parametrize solutions of genus 0 Diophantine equations using integer-valued polynomials, extending previous results on rational parametrizations.

Findings

Solutions can be parametrized by a single triple of integer-valued polynomials.

Parametrization covers all solutions except those at singular points.

Explicit construction of the parametrization is provided.

Abstract

Let f in 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 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.