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Let K be an algebraic function field of characteristic p > 2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K1, possibly equal to C, and elements u, x such that u is transcendental over K1, x is algebraic over C(u) and K = K1(u, x). Then the Diophantine problem of K is undecidable. Let G be an algebraic function field in one variable whose constant field is algebraic over a finite field and is not algebraically closed. Then for any prime p of G, the set of elements of G integral at p is Diophantine over G.

Hilbert’s Tenth Problem for algebraic function fields over infinite fields of constants of positive characteristic