On relative computability for curves

TL;DR

This paper explores the relative decidability of the finiteness problem for rational solutions to Diophantine equations, proposing that it may be decidable in contrast to the integer case, and discusses a rational version of a classical conjecture.

Contribution

It introduces a rational analogue of the Matiyasevich-Davis-Putnam conjecture, suggesting a different decidability status for rational solutions.

Findings

Proposes a suspicion that the finiteness problem is relatively decidable for rational solutions.

Formulates a rational version of the M-D-P conjecture.

Discusses implications for the decidability of Diophantine equations.

Abstract

We discuss a rational version of a conjecture of Matiyasevich, Davis, and Putnam on the relative decidability of the finiteness problem for Diophantine equations with respect to the existence problem. We formulate a suspicion that for rational solutions, the finiteness problem should be relatively decidable in contrast to the M-D-P conjecture for integer solutions.