All Together Now (with Universal Quantifiers)

numbers is undecidable. J. Robinson used quadratic forms and Hasse-Minkwoski Theorem to prove her result.

In a 2007 paper G. Cornelissen and K. Zahidi analyzed J. Robinson’s formula and showed that it can be converted to a formula of the form (V3V3)(F = 0) where the V-quantiflers run over a total of 8 variables, and where F is a polynomial. In 2008 Poonen [35] produced an improvement of the first-order definition of integers over Q. He showed that Z is definable over Q using just two universal quantifiers in a V3-formula. B. Poonen used quadratic forms, quaternions and the Hasse Norm Principle. His definition of Z over Q is simple enough to be reproduced here: the set Z equals the set of t e Q for which the following formula is true over Q:

Starting with B. Poonen’s results, J. Koenigsmann further reduced the number of quantifiers to one in [19]. As we pointed out above, this result could very well be the optimal one, since Z probably does not have a purely existential definition over Q. In the same paper, Koenigsmann showed that Z has a purely universal definition over Q or alternatively, the set of non-integers has a Diophantine definition over Q. We will return to the issue of definitions using all of the quantifiers in the sections below concerning finite and infinite algebraic extensions.

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