Last year I blogged about an argument of Bjorn Poonen, which shows that Hilbert’s tenth problem has a negative solution over the ring of integers O_K of a number field K whenever there exists an elliptic curve E/Q such that E(Q) and E(K) both have rank 1. That is: there’s no algorithm that tells you whether a given polynomial equation over O_K is solvable. The idea is that under these circumstances one can construct a Diophantine model for Z inside O_K; one already knows (by Matijasevic, Robinson, etc.) that no algorithm can determine whether a polynomial equation over Z has a solution, and the same property is now inherited by the ring O_K.
The necessary fact about existence of low-rank elliptic curves over number fields (actually, not quite the fact Poonen asked for but something weaker that suffices) has now been proven, subject to a hypothesis on the finiteness of Sha, by Mazur and Rubin: see Theorems 1.11 and 1.12. So, if you believe Sha to be finite, you believe that Hilbert’s tenth problem has a negative answer for the ring of integers of every number field.
The result of Mazur and Rubin is actually much more substantial than the corollary I mention here, giving for instance quite strong lower bounds on the number of twists of an elliptic curve E with specified 2-Selmer rank. But I haven’t studied the argument sufficiently to say anything serious about what’s inside.
I recently returned from the “Spaces of curves and their interaction with Diophantine problems” conference at Columbia, where Janos Kollar discussed the question: Which subsets S of C(t) are Diophantine? That is, which have the property that they can be written as the set of s in C(t) such that for some polynomial f in k+1 variables with coefficients in C(t). Kollar explained how to prove that the polynomial ring C[t] is not Diophantine in C(t). The idea is to show that any “sufficiently large” Diophantine subset S of C(t) contains functions whose denominators are essentially arbitrary; more precisely (but not completely precisely!) if X in Sym^d P^1 is the locus of degree-d denominators of elements of S, the Zariski closure of X needs to be — well, it doesn’t have to contain all degree-d polynomials, but it has to contain a set of the form as F,G range over polynomials of degrees s,t with s+rt = d. In particular, it’s not possible for the denominator to be identically 1, as would be the case if S were C[t]. In fact, this argument shows that no finitely generated C-subalgebra of C(t) is Diophantine over C[t].
Open question: is the localization of C(t) at t Diophantine over C(t)?
Update: When I first posted this I didn’t notice that Kollar’s result is already out, in the new journal Algebra and Number Theory, so you can go to the source for more details. ANT, by the way, is a free electronically distributed journal with a terrific editorial board, and I highly recommend submitting there.