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Badoureau (1879)

Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale

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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