Diophantine subsets of function fields of curves

TL;DR

This paper investigates diophantine subsets within function fields of curves, revealing they are either very small or very large, with implications for polynomial rings over rational function fields.

Contribution

It demonstrates that diophantine subsets in function fields are dichotomous and shows that polynomial rings are not diophantine in many cases.

Findings

Diophantine subsets are either very small or very large.

Polynomial rings are not diophantine in many fields.

Implications for the structure of diophantine sets in function fields.

Abstract

We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials $k[t]$ is a not a diophantine subset of the field of rational functions $k(t)$ for many fields $k$.