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

Authors:

Institution:

(1) Mathematisch Instituut, Universiteit Utrecht, Utrecht, Nederland;(2) Department of Mathematics, East Carolina University, Greenville, U.S.A.

Abstract:

Julia Robinson gave a first-order definition of the set of integers $\mathbb{Z}$ in the rational numbers $\mathbb{Q}$ by a formula (∀∃∀∃)(F = 0) where the ∀-quantifiers run over a total of 8 variables and F is polynomial. We show that for a large class of number fields, not including $\mathbb{Q}$ , for every ε > 0 there exist a set of primes $\mathcal{S}$ of natural density exceeding 1 − ε such that $\mathbb{Z}$ can be defined as a subset of the “large” subring

$\left\{ {x \in K:{\text{ord}}_p x \ge 0,\quad \forall \mathfrak{p} \notin \mathcal{S}} \right\}$

of K by a formula where there is only one ∀-quantifier. In the case of $\mathbb{Q}$ , we will need two quantifiers. We also show that in some cases one can define a subfield of a number field using just one universal quantifier. Bibliography: 18 titles. Dedicated to Yuri Matiyasevich on the occasion o his 60th birthday. ... the undecidable poem “B Петербрге мы сойдемся снова” ... (18]) Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 199–223.

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