A Note on Hilbert's ``Geometric'' Tenth Problem

This paper explores undecidability in theories of positive characteristic function fields in the “geometric” language of rings L F = {0, 1,+, ·, F} , with a unary predicate F for nonconstant elements. We indicate how to generalise existing machinery to prove the undecidability of the ∀ 1 ∃ + -L F theory (without parameters) of any function field of a curve over an algebraic extension of F p , not algebraically closed. We discuss the problem (and its geometric implications) further in this context too. MSC Classification: 03B25 , 12L05