A Diophantine definition of the constants in $\mathbb{Q}(z)$

TL;DR

This paper explores whether the rational numbers can be defined using Diophantine equations within the field of rational functions over the rationals, providing a conditional positive answer based on conjectures related to elliptic surfaces.

Contribution

It offers a conditional proof that $\

Findings

Conditional proof of Diophantine definability of $\\mathbb{Q}$ in $\\mathbb{Q}(z)$

Relies on standard conjectures on elliptic surfaces

Advances understanding of Diophantine problems in function fields

Abstract

It is an old problem in the area of Diophantine definability to determine whether $\mathbb{Q}$ is Diophantine in $\mathbb{Q}(z)$. We provide a positive answer conditional on two standard conjectures on elliptic surfaces.