Diophantine Sets over Polynomial Rings and Hilbert's Tenth Problem for Function Fields

In 1900, the German mathematician David Hilbert proposed a list of 23 unsolved mathematical problems. In his Tenth Problem, he asked to find an algorithm to decide whether or not a given diophantine equation has a solution (in integers). Hilbert's Tenth Problem has a negative solution, in the sense that such an algorithm does not exist. This was proven in 1970 by Y. Matiyasevich, building on earlier work by M. Davis, H. Putnam and J. Robinson. Actually, this result was the consequence of something much stronger: the equivalence of recursively enumerable and diophantine sets (we will refer to this result as "DPRM"). The first new result in the thesis is about Hilbert's Tenth Problem for function fields of curves over valued fields in characteristic zero. Under some conditions on the curve and the valuation, we have undecidability for diophantine equations over the function field of the curve. One interesting new case are function fields of curves over formal Laurent series. The proof relies on the method with two elliptic curves as developed by K. H. Kim and F. Roush and generalised by K. Eisenträger. Additionally, the proof uses the theory quadratic forms and valuations. And especially for non-rational function fields there is some algebraic geometry coming in. The second type of results establishes the equivalence of recursively enumerable and diophantine sets in certain polynomial rings. The most important is the one-variable polynomial ring over a finite field. This is the first generalisation of DPRM in positive characteristic. My proof uses the structure of finite fields and in particular the properties of cyclotomic polynomials. In the last chapter, this result for polynomials over finite fields is generalised to polynomials over recursive algebraic extensions of a finite field. For these rings we don't have a good definition of "recursively enumerable" set, therefore we consider sets which are recursively enumerable for every recursive presentation. We show that these are exactly the diophantine sets. In addition to infinite extensions of finite fields, we also show the analogous result for polynomials over a ring of integers in a recursive totally real algebraic extension of the rationals. This generalises results by J. Denef and K. Zahidi.