Diophantine Maps

Abstract

We formalize the reduction of Hilbert's tenth problem from one ring to another via the definition of Diophantine maps. We prove properties of those maps and relate them to recursive functions. We take a look at the proofs that Hilbert's tenth problem over \C(t_1,t_2) and \F_q(T) have a negative answer. Last we prove some new results for Hilbert's tenth problem over the twisted polynomial ring and over the ring of differential polynomials and some variants of both.