Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers

TL;DR

This paper establishes that in a global field, rings of $S$-integers can be universally defined with 10 quantifiers, and intersections of valuation rings can be existentially defined with 3 quantifiers, advancing definability results in number theory.

Contribution

It provides the first known universal definition of rings of $S$-integers in global fields using only 10 quantifiers and an existential definition of valuation ring intersections with 3 quantifiers.

Findings

Rings of $S$-integers are universally definable with 10 quantifiers.

Finite intersections of valuation rings are existentially definable with 3 quantifiers.

The results improve understanding of definability in number fields.

Abstract

We show that for a global field $K$, every ring of $S$-integers has a universal first-order definition in $K$ with $10$ quantifiers. We also give a proof that every finite intersection of valuation rings of $K$ has an existential first-order definition in $K$ with $3$ quantifiers.