Division-ample sets and the Diophantine problem for rings of integers
Gunther Cornelissen, Thanases Pheidas, Karim Zahidi
Abstract
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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