Hilbert's tenth problem and arithmetic geometry

In this dissertation we study generalizations of Hilbert's Tenth Problem. The original problem is the following: Is there a uniform algorithm that decides, given a multivariate polynomial equation with coefficients in the ring $Z$ of integers, whether the equation has a solution over the integers? Matijasevich proved that no such algorithm exists, i.e. that Hilbert's Tenth Problem is undecidable. Since Matijasevich's result, various analogues of this problem have been studied by asking the same question as above for multivariate polynomial equations with coefficients and solutions over some other commutative ring R. The methods that are used for proving undecidability of these generalizations come from number theory and arithmetic geometry.; In this dissertation we prove the undecidability of Hilbert's Tenth Problem for several new rings and fields.; We prove that Hilbert's Tenth Problem for function fields of surfaces over an algebraically closed field of characteristic zero is undecidable. This generalizes a result by Kim and Roush from 1992.; Shlapentokh proved that the problem for algebraic function fields over possibly infinite constant fields of characteristic p > 2 is undecidable. In this dissertation we solve the analogous problem for function fields of characteristic 2. In particular this shows that Hilbert's Tenth Problem for any global field of positive characteristic is undecidable.; Let K be the perfect closure of a global field of characteristic p > 2. We prove that the set of elements which are integral at a given prune $p$ of K is diophantine. This is one of two steps needed to prove undecidability for K.; Diophantine undecidability for rings of integers of number fields is known for only a restricted class. We prove that using integral points on non-split tori in the same way earlier authors used Pell equations will not yield new results, so their methods cannot be generalized.; Finally, we prove that Hilbert's Tenth Problem is undecidable for localizations of finitely generated $Z$-algebras that have characteristic n > 0 and that have infinitely many elements. This generalizes the results for function fields of characteristic p > 0. We also prove a conditional result in characteristic zero.