Defining the integers in large rings of number fields using one universal quantifier

TL;DR

The paper extends Julia Robinson's definability of integers to large subrings of number fields, using a formula with fewer quantifiers, applicable to most primes in these fields.

Contribution

It introduces a new definability result for integers in large subrings of number fields with a simplified quantifier structure, broadening previous work beyond the rationals.

Findings

Defines integers in large subrings using a single universal quantifier

Applicable to a broad class of number fields excluding Q

Achieves a definability with fewer quantifiers than prior methods

Abstract

Julia Robinson has given a first-order definition of the rational integers $\mathbb Z$ in the rational numbers $\mathbb Q$ by a formula $(\forall \exists \forall \exists)(F=0)$ where the $\forall$-quantifiers run over a total of 8 variables, and where F is a polynomial. We show that for a large class of number fields, not including $\mathbb Q$, for every $\epsilon>0$, there exists a set of primes $\cal S$ of natural density exceeding $1-\epsilon$, such that $\mathbb Z$ can be defined as a subset of the ``large'' subring $$\{x \in K : \ord_{\mathfrak p}x >0, \forall \mathfrak p \not \in \cal S \}$$ of K by a formula of the form $(\exists \forall \exists)(F=0)$ where there is only one $\forall$-quantifier, and where F is a polynomial.