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Let  k   be a field of characteristic zero,  V   a smooth,  positive-dimensional, quasiprojective variety over  k , and  Q   a  nonempty divisor on  V . Let  K   be the function field of   V , and  A  ⊂  K   the semilocal ring of  Q .

Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real and p -adic fields