Undecidable existential theories of polynomial rings and function fields

Abstract

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 ∊ C(t)};(b)any polynomial ring over an integral domain of constants, in the lansruage of rings augmented by a symbol for the nonconstant polynomials.