Having resolved the issues of Diophantine decidability over global function fields, we turn our attention to Diophantine definability over these fields. Our goal for this chapter is to produce vertical and horizontal definability results for “large” rings of functions, as we did for “large” number rings. The original results discussed in this chapter can be found in [93], [100] and [103].

We start with a function field version of the weak vertical method.

The weak vertical method revisited

In this section we revisit the weak vertical method and adjust it for function fields. As will be seen below, very little “adjusting” will be required.

Theorem 10.1.1. The weak vertical method for function fieldsLet K/L be a finite separable extension of function fields over finite fields of constants and let KN be the normal closure of K over L. Let {ω1 = 1, …, ωK} be a basis of K over L. Let Z ∈ K. Further, let V be a finite set of primes of K satisfying the following conditions.

1. Each prime of V is unramified over L and is the only K-factor of the prime below it in L.

2. Z is integral at all the primes of V.

3. For each p ∈ V there exists b(p) ∈ CL (the constant field of L) such that

4. {ω1, …, ωk} is a local integral basis with respect to every prime of V.

5. | V |> ka (Ω)HKN (Z), where a(Ω) is a constant defined as in Lemma 9.1.3.