Undecidable existential theories of polynomial rings and function fields

We prove the undecidability of the positive existential theories of: (a) the field C(t): in the language of rings augmented by a constant t and a symbol for the set of derivatives D, D = {x' : x is an element of C(t)}; (b) any polynomial ring over an integral domain of constants, in the language of rings augmented by a symbol for the nonconstant polynomials. Introduction: Hilbert's tenth problem, concerning the algorithmic treatment of the general diophantine problem, was resolved negatively for the ring of integers in [Matijasevich], although it remains open in general for the ring of integers in, a number held. In the past two decades considerable attention has been paid to the analogous problem over polynomial rings and fields of rational functions: cf. [Denef, 1], [Denef,2], [KR,1], [KR,2] and [Pheidas]. To date all the results in this area have been negative, that is, in all cases which have been resolved the positive existential theories have been shown to be undecidable. For example, it is known that if A(t) is a polynomial ring over an integral domain A then its positive existential theory in the language L = {0, 1, t, divided by, .} is undecidable. Similar negative results have been obtained for certain fields of rational functions, notably R(t) with R a formally real field field, and the field C(t(1), t(2)) of rational functions in two variables with complex coefficients.