Characterizing integers among rational numbers with a universal-existential formula

Abstract

We prove that Z in definable in Q by a formula with 2 universal quantifiers followed by 7 existential quantifiers. It follows that there is no algorithm for deciding, given an algebraic family of Q-morphisms, whether there exists one that is surjective on rational points. We also give a formula, again with universal quantifiers followed by existential quantifiers, that in any number field defines the ring of integers.

6 pages

Cited by in corpus (4)

• Bounds of some real (complex) solution of a finite system of polynomial equations with rational coefficients
• Irreducibility of Polynomials over Global Fields is Diophantine
• A p-adic analogue of Siegel's Theorem on sums of squares
• Universally defining $\mathbb{Z}$ in $\mathbb{Q}$ with $10$ quantifiers