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

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 $\mathcal{S}$ 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.

Substantial changes in Theorems 1 and 2 and their proofs. Two new theorems (3 and 4)