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A failure of quantifier elimination.

Angus Macintyre, David Marker (1997)

Revista Matemática de la Universidad Complutense de Madrid

We show that log is needed to eliminate quantifiers in the theory of the real numbers with restricted analytic functions and exponentiation.

Definability within structures related to Pascal’s triangle modulo an integer

Alexis Bès, Ivan Korec (1998)

Fundamenta Mathematicae

Let Sq denote the set of squares, and let S Q n be the squaring function restricted to powers of n; let ⊥ denote the coprimeness relation. Let B n ( x , y ) = ( x + y x ) M O D n . For every integer n ≥ 2 addition and multiplication are definable in the structures ⟨ℕ; Bn,⊥⟩ and ⟨ℕ; Bn,Sq⟩; thus their elementary theories are undecidable. On the other hand, for every prime p the elementary theory of ⟨ℕ; Bp,SQp⟩ is decidable.

Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Laurent Moret-Bailly, Alexandra Shlapentokh (2009)

Annales de l’institut Fourier

Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K , not equal to K . We prove the following undecidability results for R : if K is recursive, then Hilbert’s Tenth Problem is undecidable in R . In general, there exist x 1 , ... , x n R such that there is no algorithm to tell whether a polynomial equation with coefficients in ( x 1 , ... , x n ) has solutions in R .

Division-ample sets and the Diophantine problem for rings of integers

Gunther Cornelissen, Thanases Pheidas, Karim Zahidi (2005)

Journal de Théorie des Nombres de Bordeaux

We prove that Hilbert’s Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K ). We relate division-ample sets to arithmetic of abelian varieties.

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