Diophantine Subsets of Function Fields of Curves

Fi(r, y1, . . . , yn) = 0 ∀i has a solution (y1, . . . , yn) ∈ R iff r ∈ D. Equivalently, if there is a (possibly reducible) algebraic variety XR over R and a morphism π : XR → A 1 R such that D = π(XR(R)). In this situation we call dioph(XR, π) := π(XR(R)) ⊂ R the diophantine set corresponding to XR and π. A characterization of diophantine subsets of Z was completed in connection with Hilbert’s 10th problem, but a description of diophantine subsets of Q is still not known. In particular, it is not known if Z is a diophantine subset of Q or not. (See [Poo03] or the volume [DLPVG00] for surveys and many recent results.) In this paper we consider analogous questions where R = k(t) is a function field of one variable and k is an uncountable large field of characteristic 0. That is, for any k-variety Y with a smooth k-point, Y (k) is Zariski dense. Examples of uncountable large fields are