In 1970, Yuri Matiyasevich, building on work by Martin Davis, Hilary Putnam and Julia Robinson, proved that Hilbert's Tenth Problem over the integers is undecidable. The analogue of Hilbert’s Tenth Problem over the rationals, and over number fields in general, remains open. A diophantine definition of the integers over the rationals, together with a standard reduction argument, would show that Hilbert’s Tenth Problem over the rationals is undecidable. A diophantine definition of the integers over the rationals seems out of reach right now, but one can also consider the problem of defining the integers inside the rationals or number fields with a first-order formula. Julia Robinson showed that this is possible, and we will discuss her proof as well as recent improvements and extensions of her result.

We will also discuss examples of sets that can be shown to be diophantine in number fields and in global function fields, such as the set of quadratic non-norms and the set of non-squares.

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