The questions and problems raised by the solution of Hilbert's Tenth Problem extend to many other objects besides the global fields which are the subject of this book. In particular, the questions we have raised are just as relevant for infinite algebraic extensions of global fields, for fields of positive characteristic and of transcendence degree greater than 1, and for function fields of characteristic 0. A detailed and substantial discussion of existential definability and decidability over these objects is beyond the scope of this book, but in this chapter we will briefly survey extensions of Hilbert's Tenth Problem to some objects mentioned above, so that an interested reader can be directed to the original sources. Before proceeding, we would like to note that many detailed surveys of the subjects discussed in this chapter can be found in [20].

Function fields of positive characteristic and of higher transcendence degree or over infinite fields of constants

The most general result concerning rational function fields of positive characteristic was obtained by H. K. Kim and F. W. Roush in [42]. They proved the following proposition.

Theorem 13.1.1.Let K be a rational function field over a field C of constants of characteristic p > 0. Assume that C does not contain the algebraic closure of a finite field.