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.
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