For a ring R, Hilbert’s Tenth Problem is the set HTP(R) of polynomials f ∈ R[X₁,X₂,…] for which f=0 has a solution in R. Matiyasevich, completing work of Davis, Putnam, and Robinson, showed that HTP(Z) is Turing-equivalent to the Halting Problem. The Turing degree of HTP(Q) remains unknown. Here we consider the problem for subrings of Q. One places a natural topology on the space of such subrings, which is homeomorphic to Cantor space. This allows consideration of measure theory and also Baire category theory. We prove, among other things, that HTP(Q) computes the Halting Problem if and only if HTP(R) computes it for a nonmeager set of subrings R.

We will consider several number-theoretic questions which arise from computable model theory. One of these, recently solved by Poonen, Schoutens, Shlapentokh, and the speaker, involves attempting to “embed” a graph into a field: given a graph, one wishes to construct a field with the exact same computable-model-theoretic properties as the graph. (For instance, the automorphisms of the graph should correspond to the automorphisms of the field, by a bijective functorial correspondence which preserves the Turing degree of each automorphism.) Another arises out of consideration of Hilbert’s Tenth Problem for subrings of the rationals: we ask for subrings in which Hilbert’s Tenth Problem is no harder than it is for the rationals themselves. This is known for semilocal subrings, and Eisenträger, Park, Shlapentokh and the speaker have shown that it holds for certain non-semilocal subrings as well, but it remains open whether one can invert “very few” primes and still have it hold. We will explain this problem and discuss the number-theoretic question which arises out of it.

In the year 2003 I first heard of the notion of extremal valued fields when Yuri Ershov gave a talk at a conference in Teheran. He proved that algebraically complete discretely valued fields are extremal. However, the proof contained a mistake, and it turned out in 2009 through an observation by Sergej Starchenko that Ershov’s original definition leads to all extremal fields being algebraically closed. In joint work with Salih Durhan (formerly Azgin) and Florian Pop, we chose a more appropriate definition and then characterized extremal valued fields in several important cases.

We call a valued field $(K,v)$ extremal if for all natural numbers n and all polynomials $f$ in $K[X_1,…,X_n]$, the set of those $f(a_1,…,a_n)$ with $a_1,…,a_n$ in the valuation ring has a maximum (which is allowed to be infinity, which is the case if $f$ has a zero in the valuation ring). This is such a natural property of valued fields that it is in fact surprising that it has apparently not been studied much earlier. It is also an important property because Ershov’s original statement is true under the revised definition, which implies that in particular all Laurent Series Fields over finite fields are extremal. As it is a deep open problem whether these fields have a decidable elementary theory and as we are therefore looking for complete recursive axiomatizations, it is important to know the elementary properties of them well. That these fields are extremal seems to be an important ingredient in the determination of their structure theory, which in turn is an essential tool in the proof of model theoretic properties.

Further, it came to us as a surprise that extremality is closely connected with Pop’s notion of “large fields”. Also the notion of tame valued fields plays a crucial role in the characterization of extremal fields. A valued field $K$ with algebraic closure $K^{acl}$ is tame if it is henselian and the ramification field of the extension $K^{acl}|K$ coincides with the algebraic closure.

In my talk I will introduce the above notions, try to explain their meaning and importance also to the non-expert, and discuss in detail what is known about extremal fields and how the properties of large and of tame fields appear in the proofs of the characterizations we give. Finally, I will present some challenging open problems, the solution of which may have an impact on the above mentioned problem for Laurent Series Fields over finite fields.