Diophantine Undecidability over Algebraic Function Fields over Finite Fields of Constants Original Research Article

We show that Diophantine problem (otherwise known as Hilbertʹs Tenth Problem) is undecidable over the fields of algebraic functions over the finite fields of constants of characteristic greater than two. This is the first example of Diophantine undecidability over any algebraic field. We also show that the Diophantine class of a holomorphy ring of an above mentioned algebraic function field does not change if the set of primes at which the functions of the ring are allowed to have poles is changed by adding or removing of finitely many primes.