Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a multivariate polynomial equation with integer coefficients, whether it has a solution over the integers. In 1970 Matiyasevich, building on work by Davis, Putnam and Robinson, proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. In this talk we will consider generalizations of Hilbert's Tenth Problem and Mazur's conjectures for large subrings of number fields. We will show that Hilbert's Tenth Problem is undecidable for large complementary subrings of number fields and that the analogues of Mazur's conjectures do not hold in these rings

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