# Blog Archives

# Topic Archive: field theory

# The logical complexity of Schanuel’s Conjecture

In its most natural form Schanuel’s Conjecture is a $\Pi_1^1$-statement. We will show that there is an equivalent $\Pi^0_3$-statement. They key idea is a result of Jonathan Kirby showing that, if Schanuel’s Conjecture is false, then there are canonical counterexamples. Most of my lecture will describe Kirby’s work.

# Factoring polynomials and finding roots

We will use computability theory to investigate the difficulty of two standard questions from algebra. First, given a field *F* and a polynomial *p∈ F[X]*, does the equation *p=0* have a solution in *F*? And second, is *p(X)* irreducible in *F[X]*? Obviously these two questions are related, but it is not immediately clear which one is more difficult to answer. Indeed, it is not clear how one should measure the “difficulty” of answering these questions.

To address this problem, we will consider *computable fields*, which are countable fields in which we have algorithms for listing all the field elements and for the basic field operations. It has been known since 1960 that in this context, the two questions are equivalent under the standard notion of *Turing reducibility*: an algorithm for answering either question can be used to give an algorithm for answering the other one. However, computability theory also provides more precise measures of difficulty, and using one of these measures, known as *1-reducibility*, we will show that one of the two questions can definitively be said to be more difficult in general than the other. To find out which is which, come to the talk!

# Computability problems in number theory

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.

# Lengths of roots of polynomials over *k((G))*

Mourgues and Ressayre showed that any real closed field $R$ can be mapped isomorphically onto a truncation-closed subfield of the Hahn field $k((G))$, where $G$ is the natural value group of $R$ and $k$ is the residue field. If we fix a section of the residue field and a well ordering < of $R$, then the procedure of Mourgues and Ressayre yields a canonical section of $G$ and a unique embedding $d: R$ → $k((G))$. Julia Knight and I believed we had shown that for a real closed field $R$ with a well ordering < of type ω, the series in $d(R)$ have length less than ω^{ωω}, but we found a mistake in our proof. We needed a better understanding of what happens to lengths under root-taking. In this talk, we give a partial answer, which allows us to prove the original statement and generalize it. We make use of unpublished notes of Starchenko on the Newton-Puiseux method for taking roots of polynomials.

# Finding definable henselian valuations

(Joint work with Jochen Koenigsmann.) There has been a lot of recent progress in the area of definable henselian valuations. Here, a valuation is called *definable* if its valuation ring is a first-order definable subset of the field in the language of rings. Applications of results concerning definable henselian valuations typically include showing decidability of the theory of a field or facts about its absolute Galois group.

We study the question of which henselian fields admit definable henselian valuations with and without parameters. In equicharacteristic 0, we give a complete characterization of henselian fields admitting parameter-definable (non-trivial) henselian valuations. We also give a partial characterization result for the parameter-free case.

# Towards a model theory of Zariski-Riemann spaces of valuations

Zariski introduced the space of all valuations on a given field *K* and named it after his mentor the `Riemann manifold’. This terminology is justified because of the following two facts he proved about it: (1) one can define a (quasi-)compact topology on this space (and we honor him embracingly by calling it the Zariski-Riemann space), and (2) if *K* is the function field of a curve, then this space is isomorphic to a non-singular curve with the same function field. Hence (2) gives resolution of singularities in dimension one, and he then used fact (1) to show the same in dimension two. Abhyankar then followed suit by proving that in dimension two, any point (=valuation) in this space is attainable by blowing ups, but his student Shannon showed that this failed in dimension three and higher. This latter negative result somehow ended the Zariski project of using the space of all valuations to prove resolution of singularities in higher dimensions.

However, there is a resurgence of this space and its use in resolution of singularities during the last decades, and often, these results are combined with model-theoretic techniques (Kuhlmann, Knaf, Pop, Cutkosky, Cossart, Piltant, Teissier, Scanlon,…). However, the model-theoretic setting always departs from Robinson’s point of view of a valued field: a field together with a valuation. However, the Zariski-Riemann space talks about not just one valuation, but all, so that we need a new framework. I will present some preliminary remarks of how this could be done using either a simple-minded one-sorted language or a more sophisticated two-sorted language. As a simple application of the one-sorted case, I will reprove fact (1) by simply relating it to the compactness of the Stone space of types.

# Mordell-Lang and Manin-Mumford in positive characteristic, revisited

We give a reduction of function field Mordell-Lang to function field Manin-Mumford, in positive characteristic. The upshot is another account of or proof of function field Mordell-Lang in positive characteristic, avoiding the recourse to difficult results on Zariski geometries.

(This work is joint with Benoist and Bouscaren.)

# Extremal fields, tame fields, large fields

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.

# The theory of fields is complete for isomorphisms

We give a highly effective coding of countable graphs into countable fields. For each countable graph $G$, we build a countable field $F(G)$, uniformly effectively from an arbitrary presentation of $G$. There is a uniform effective method of recovering the graph $G$ from the field $F(G)$. Moreover, each isomorphism $g$ from $G$ onto any $G’$ may be turned into an isomorphism $F(g)$ from $F(G)$ onto $F(G’)$, again by a uniform effective method so that $F(g)$ is computable from $g$. Likewise, an isomorphism $f$ from $F(G)$ onto any $F(G’)$ may be turned back into an isomorphism $g$ with $F(g)=f$. Not every field $F$ isomorphic to $F(G)$ is actually of the form $F(G’)$, but for every such $F$, there is a graph $G’$ isomorphic to $G$ and an isomorphism $f$ from $F$ onto $F(G’)$, both computable in $F$.

It follows that many computable-model-theoretic properties which hold of some graph $G$ will carry over to the field $F(G)$, including spectra, categoricity spectra, automorphism spectra, computable dimension, and spectra of relations on the graph. By previous work of Hirschfeldt, Khoussainov, Shore, and Slinko, all of these properties can be transferred from any other countable, automorphically nontrivial structure to a graph (and then to various other standard classes of structures), so our result may be viewed as saying that, like these other classes, fields are complete for such properties.

This work is properly joint with Jennifer Park, Bjorn Poonen, Hans Schoutens, and Alexandra Shlapentokh. The slides for the talk are available here.

# Logarithmic-Exponential Series

I will survey some old work of van den Dries, Macintyre and myself. We construct an algebraic nonstandard model of the theory of the real exponential field. There is a natural derivation on the LE-series which is compatible with the exponential and the archimedean valuation.

# LE-Series

In this expository talk I will discuss the construction of the field of LE-series after van den Dries, Macintyre, and Marker. The field of LE-series is an ordered differential field extending the field of real Laurent series which also has a well-behaved exponential function. The field of LE-series is closed under a host of operations, in particular it is closed under formal integration as well as compositional inverse (once composition has been properly interpreted). As such this field may be viewed, at least conjecturally, as providing a universal domain for ordered differential algebra as witnessed in Hardy fields.

# Why model-theorists shouldn’t think that ACF is easy

We all learned that stability theory derived many of its ideas from what happens in ACF, where everything is nice and easy. After all ACF has quantifier elimination and is strongly minimal, decidable, superstable, uncountably categorical, etc. However, my own struggles with ACF have humbled my opinion about it: it is an awfully rich theory that encodes way more than our current knowledge. I will discuss some examples showing how “difficult” ACF is: Grothendieck ring, isomorphism problem, set-theoretic intersection problem. Oddly enough, RCF seems to not have any of these problems. It is perhaps my ignorance, but I have come to think of RCF as much easier. Well, all, of course, is a matter of taste.

# Independent sets in computable free groups and fields

We consider maximal independent sets within various sorts of groups and fields freely generated by countably many generators. The simplest example is the free divisible abelian group, which is just an infinite-dimensional rational vector space. As one moves up to free abelian groups, free groups, and “free fields” (i.e. purely transcendental field extensions), maximal independent sets and independent generating sets both become more complicated, from the point of view of computable model theory, but sometimes in unpredictable ways, and certain questions remain open. We present the topic partly for its own sake, but also with the intention of introducing the techniques of computable model theory and illustrating some of its possible uses for an audience to which it may be unfamiliar.

This is joint work with Charles McCoy.

# An algebraic characterization of recursively saturated real closed fields

We (with D’Aquino and Kuhlmann) give a valuation theoretic characterization for a real closed field to be recursively saturated. Previously, Kuhlmann, Kuhlmann, Marshall, and Zekavat gave such a characterization for kappa-saturation, for all infinite cardinals kappa. Our result extends the characterization for a divisible ordered abelian group to be recursively saturated found in some unpublished work of Harnik and Ressayre.