Universally and existentially definable subsets of global fields

TL;DR

This paper demonstrates that certain algebraic subsets of global fields, such as rings of $S$-integers and non-squares, are definable using first-order logic, extending previous results and providing new proofs.

Contribution

It extends the class of algebraic sets known to be first-order definable in global fields, including rings of $S$-integers and non-squares, and offers new proofs for existing theorems.

Findings

Rings of $S$-integers are universally definable in global function fields of odd characteristic.

The set of non-squares in a global field of characteristic not 2 is diophantine.

The set of pairs where one element is not a norm in a quadratic extension is diophantine over the field.

Abstract

We show that rings of $S$-integers of a global function field $K$ of odd characteristic are first-order universally definable in $K$. This extends work of Koenigsmann and Park who showed the same for $\mathbb{Z}$ in $\mathbb{Q}$ and the ring of integers in a number field, respectively. We also give another proof of a theorem of Poonen and show that the set of non-squares in a global field of characteristic $\neq 2$ is diophantine. Finally, we show that the set of pairs $(x,y)$ in $(K^{\times})^2$ such that $x$ is not a norm in $K(\sqrt{y})$ is diophantine over $K$ for any global field $K$ of characteristic $\neq 2$.