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In this article we outline the methods that are used to prove undecidability of Hilbert's Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of dimension at least 2.
Hilbert's Tenth Problem in its original form was to find an algorithm to decide, given a polynomial equation f(x1,…, xn) = 0 with coefficients in the ring ℤ of integers, whether it has a solution with x1,…, xn ∈ ℤ. Matijasevič ([Mat70]), based on work by Davis, Putnam and Robinson ([DPR61]), proved that no such algorithm exists, i.e. Hilbert's Tenth Problem is undecidable. Since then, analogues of this problem have been studied by asking the same question for polynomial equations with coefficients and solutions in other commutative rings R. We will refer to this as Hilbert's Tenth Problem over R. Perhaps the most important unsolved question in this area is the case R = ℚ. There has been recent progress by Poonen ([Poo03]) who proved undecidability for large subrings of ℚ. The function field analogue, namely Hilbert's Tenth Problem for the function field k of a curve over a finite field, is undecidable.
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