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

Cornelissen, Gunther; Shlapentokh, Alexandra

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

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Description

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 $ε>0$, there exists a set of primes $\cal S$ of natural density exceeding $1-ε$, 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)