Talk at the model theory seminar, Oxford 2016

In the year 2003 Yuri Ershov gave a talk at a conference in Teheran on his notion of ``extremal valued fields''. 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 values {vf(a_1,...,a_n) | a_1,...,a_n in the valuation ring} has a maximum (which is allowed to be infinity, attained 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 could 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.

The notion of "tame valued field" and their model theoretic properties play a crucial role in the characterization of extremal fields. A valued field K with separable-algebraic closure K^sep is tame if it is henselian and the ramification field of the extension K^sep|K coincides with the algebraic closure. Open problems in the classification of extremal fields have recently led to new insights about elementary equivalence of tame fields in the unequal characteristic case. This led to a follow-up paper. Major suggestions from the referee were worked out jointly with Sylvy Anscombe and led to stunning insights about the role of extremal fields as ``atoms'' from which all aleph_1-saturated valued fields are pieced together.

Talk at the conference Ordered Algebraic Structures and Related Topics, CIRM, Luminy (France), October 12-16, 2015

(joint work with Katarzyna Kuhlmann)

Which are the ordered fields in which every descending chain of nonempty closed intervals has a nonempty intersection? This is true for the reals, but are there other ordered fields that have this property? Or does it imply that the field is cut complete and hence isomorphic to the reals? The clue is that you can zoom in on a cut using a descending chain of nonempty closed intervals only if the cofinality of the left cut set is equal to the coinitiality of the right cut set; we call such cuts "symmetrical". Hence an ordered field has the above property if and only if it has no symmetrical cuts; we call such fields (and other ordered structures) "symmetrically complete". In [1] we have shown that such fields satisfy an interesting fixed point theorem; It is quite similar to Banach's fixed point theorem although the fields need not be archimedean ordered.

Saharon Shelah [3] showed in 2004 that every ordered field can be extended to a symmetrically complete field. In joint work [2] with Katarzyna and Saharon, we have given a direct construction of such fields using power series. We construct first symmetrically complete linear orderings, then symmetrically complete ordered Hahn groups having these as their index sets and archimedean components isomorphic to the reals. In fact, we also show that every symmetrically complete ordered abelian group must be of this form, hence in particular divisible. Similarly, the power series fields over the reals with symmetrically complete value groups (plus a minor additional condition) are symmetrically complete ordered fields. Conversely, every symmetrically complete ordered field is of this form, hence in particular real closed.

[1] Kuhlmann, F.-V. and Kuhlmann, K.: A common generalization of metric, ultrametric and topological fixed point theorems, Forum Math. 27 (2015), 303-327, and: Correction to A common generalization of metric, ultrametric and topological fixed point theorems,
Forum Math.**27** (2015), 329-330

alternative corrected version: (pdf file)

[2] Kuhlmann, F.-V., Kuhlmann, K. and Shelah, S.: Symmetrically complete ordered sets, abelian groups and fields, Israel J. Math. **208** (2015), 261-290

(pdf file)

[3] Shelah, S.: Quite Complete Real Closed Fields, Israel J. Math. 142 (2004), 261-272

Colloquium talk at Valladolid, Spain, 2013

During my joint work with my wife Katarzyna on particular topological spaces that appear in real algebraic geometry ("spaces of real places"), it turned out that already one of the "simplest" such spaces has a very intriguing structure with lots of self-similarities. This led us to the question whether some sort of fractality is around here. But to study fractals, one commonly uses fixed point theorems. For example, Banach's Fixed Point Theorem states that every strictly contracting function on a metric space X has a unique fixed point, that is, there is some x in X such that f(x)=x.

Unfortunately, the spaces we study are not necessarily metric, and their natural topology is not easy to handle. So we wondered whether suitable tools could be borrowed from other, better known settings. In valuation theory, metrics are replaced by ultrametrics (in which every triangle has at least two equal sides and every element in a ball is a center of that ball). The best known example is the p-adic metric. Important principles, like Hensel's Lemma for the p-adics, can be proven by means of ultrametric fixed point theorems. Working with such theorems, I had asked more than 15 years ago the following question: is there a generalization of the ultrametric fixed point theorems in a more topology-like language, that may serve as a common ground for both the metric and the ultrametric world? The answer is yes (and is surprisingly simple). I will report on a very basic fixed point theorem for that works in a minimal setting, not involving any metrics. This theorem not only covers the metric and ultrametric, but also generates a new topological fixed point theorem. The surprise coming with this theorem is that it is not the continuous, but the closed functions that are natural candidates for such theorems.

The key idea that led to the basic theorem was to use the notions "ball" and "spherically complete" from the ultrametric world but to let the balls be given by any collection of subsets you are interested in. For example, in an ordered field, you can take the balls to be the closed intervals. Then "spherically complete" means that the intersection over any chain of closed intervals (ordered by inclusion) is nonempty. From our general theorem one can derive interesting fixed point theorems for such fields. Clearly, the reals have this property, but it is a non-trivial question whether there are any other ordered fields that have it. In a paper in the Israel J. Math. in 2004, Saharon Shelah showed that there are arbitrarily large such fields, which he constructs as unions over ascending chains of ordered fields (a typical logician's approach). In very recent work, we succeeded to characterize these fields and to construct them in one step as power series fields.

As a further application of the basic theorem, fixed point theorems in lattice theory (that have been applied e.g. in computer science) can be proved. For example, the existence part of the Knaster-Tarski Theorem can be derived using the general theorem. We are now investigating how our "ball spaces" fit into these settings, and what the analogues of other results in lattice theory in the ball spaces could be.

Talk at the Conference in honour of Shreeram S. Abhyankar on occasion of his 82nd birthday, Purdue University, July 5-8, 2012

Elimination of wild ramification is an important part of local uniformization. In the particular case of an immediate valued function field F|K of transcendence degree 1 it means to find an element x in F such that F lies in the henselization of K(x). (An extension of valued fields is called immediate if value group and residue field do not change.)

This task is trivial if the characteristic of the residue field is 0, and highly nontrivial in the positive characteristic case. In order to prove a theorem of "Henselian Rationality" over suitable valued ground fields K, we need quite a bit of technical preparation, which can be seen as a continuation of Kaplansky's important paper "Maximal fields with valuations I", Duke Math. Journal 9 (1942), 303-321.

In the first step of the proof, we find the element x in the henselization F^h of F. In this step it is essential to know the degree of the extension K(x)^h|K(f(x))^h for polynomials f.

In the second step of the proof, once x is found in F^h, we try to replace it by an element y in F. Here, we need to know the degree of the extension K(x)^h|K(y)^h for arbitrary elements in K(x)^h.

This step can be seen as a special case of "dehenselization", corresponding to the notion of "decompletion" that appears in Temkin's work. Dehenselization in general states that if there is a finite extension F' of F within its henselization F^h such that F'|K admits local uniformization, then so does F|K. Whether this is true is an important open problem. The above special case and Temkin's "decompletion" seem to indicate that the problem could be solved to the affirmative.

One tool in the proofs is to use ramification theory to reduce the problem to considering Galois extensions of degree p, by extending the ground field. Then in the end, henselian rationality has to be pulled down through such ("tame") extensions. This too can be solved within the described framework where criteria are given for the degree of the extension K(x)^h|K(f(x))^h to be 1. Concretely, it can be shown that the trace of a well chosen henselian generator "upstairs" is a henselian generator "downstairs".

Talk at the SECOND INTERNATIONAL CONFERENCE AND WORKSHOP ON VALUATION THEORY , Segovia, Spain, July 18 to 29, 2011

(pdf file)

Talk in the Honorary Colloquium on Occasion of Wilfried Buchholz' 60th Birthday at the Mathematical Institute of the University of Munich, April 4 and 5, 2008:

Completeness and decidability of mathematical theories are a main subject in model theory. They have been of particular interest in model theoretic algebra. There are nice examples from field theory that by now may well be called ``classical''. Model theoretical results for algebraically closed, real closed and p-adically closed fields have found interesting applications: Hilbert's 17. Problem, Nullstellensaetze, description of positive definite polynomials and their p-adic analogues. In the year 1965 Ax and Kochen generated much interest in model theory through their proof of a correct version of Artin's Conjecture about non-trivial zeros of forms over the p-adic numbers. Since then one of the best known open problems in model theoretic algebra is whether the elementary theory of the field F

Again in 1965, another famous theorem was proved: Hironaka showed resolution of singularities for all algebraic varieties over fields of characteristic 0. Since then also for this theorem its analogue in positive characteristic has remained an open problem, in spite of all attacks from excellent algebraic geometers. Since Zariski it is known that the local version of resolution of singularities, called ``local uniformisation'', is of valuation theoretical nature. Yet it was a quite unexpected finding that t he decidability problem and the problem of local uniformisation both are based on the same valuation theretical problem: the defect. In the presence of defect, the classification of valued fields up to elementary equivalence, relative to their invariants (value group and residue field), breaks down.

Another good indication for the connection between the two problems is
the work of Macintyre and Schoutens. They show that if resolution of
singularities in positive characteristic holds, then at least the
universal elementary theory of F_{p}((t)) is decidable.

Talk in the Model Theory and Applications to Algebra and Analysis Programme at the Isaac Newton Institute for Mathematical Sciences, June 2005:

I give a survey on additive polynomials, their role in valuation theory and algebraic geometry and what we know and do not know about them. I show their connection with the defect of valued field extensions and the meaning of the defect for the model theory of valued fields and for local uniformization in positive (residue) characteristic. I mention a classification of Artin-Schreier extensions and its applications. I also talk about maximality properties of valued fields and Ershov's notion of extremal fields.

*List of open problems and references for this talk*:
(dvi file) ---
(postscript file) ---
(pdf file)

Talk in the Workshop on resolution of singularities, factorization of birational mappings, and toroidal geometry at the Banff International Research Station, December 2004:

The problem of local uniformization can be reformulated as a problem about the structure of valued function fields. I will quickly review this reformulation.

One part of the problem is elimination of ramification. This is harder in positive characteristic because there, you also have to deal with wild ramification, and the main obstacle turns out to be the defect of valued field extensions. I will give examples of non-trivial defect. Then I will show why certain valuations (called "Abhyankar valuations") always admit a positive solution, and describe the valuation theoretical theorems used for the solution.

In order to generalize these theorems, we have to learn more about the defect. A first step is to classify Artin-Schreier extensions with non-trivial defect, which I have done in a recent preprint. Based on this classification, there is joint work in progress with O. Piltant on the higher ramification groups of such extensions. Future work shall also lead to a better understanding of defect extensions generated by additive polynomials other than the Artin-Schreier polynomial.

International Congress on Nonstandard Models of Arithmetic and Analysis at the University of Pisa, June 24 to 28, 2004:

Answering a question asked at the Conference ``Logic, Algebra and Arithmetic'' in Teheran 2003, We show that every real closed (or more generally, henselian) non-archimedean ordered field L admits a proper dense subfield. This is interesting because any integer part of K will also be an integer part of L. If L is ``small'', then L|K must be algebraic. However, there are examples of trdeg L|K being any pre-assigned cardinal. In particular, if the natural valuation on L has no coarsest non-trivial coarsening, then trdeg L|K can be any countable cardinal. At the same time our construction provides a counterexample to the following conjecture:

Seminaire de Structures Algebriques Ordonnees, Univ. Paris 7, France, May 2004:

We classify Artin-Schreier extensions of valued fields with non-trivial defect according to whether they are connected with purely inseparable extensions with non-trivial defect, or not. We use this classification to show that in positive characteristic, a valued field is algebraically complete if and only if it has no proper immediate algebraic extension and every finite purely inseparable extension is defectless. This result is an important tool for the construction of algebraically complete fields. We also use the result to show that extremal fields are algebraically complete. A valued field (K,v) is called extremal if for all polynomials f in several variables the value set vf(K^n) has a maximum. Restricting this condition to certain classes of polynomials yields further interesting properties. In that way, we give characterizations of algebraically maximal and inseparably defectless fields. Finally, we give a second characterization of algebraically complete fields, in terms of their completion. As an example by Cutkosky and Piltant shows, a certain property called relative resolution may work with one type of Artin Schreier defect extensions, but not with the other. This connection with algebraic geometry has to be investigated further.

This work was strongly inspired by the first part of Francoise Delon's thesis. Some results are generalized, some others are put in a larger perspective.

Seminaire de Structures Algebriques Ordonnees, Univ. Paris 7, France, April 2004:

We classify all possible extensions of a valuation from a ground field K to a rational function field in one or several variables over K. We determine which value groups and residue fields can appear, and we show how to construct extensions having these value groups and residue fields. In particular, we give constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. One can even construct valuations on rational function fields in two variables which admit an infinite tower of degree p extensions with defect p. Such nasty valuations constitute a serious empediment for local uniformization in positive characteristic, so it is important to study their structure in detail. In the case of a rational function field K(x) in one variable, we consider the relative algebraic closure of K in the henselization of K(x) with respect to the given extension, and we show that this can be any countably generated separable-algebraic extension of K. In the ``tame case'', we show how to determine this relative algebraic closure. These methods can be applied to power series fields and to the p-adics.

Workshop and Conference on Logic, Algebra and Arithmetic at the Institute for Theoretical Physics and Mathematics, Tehran, October 2003:

A polynomial f over an infinite field K is called additive if f(a+b)=f(a)+f(b) for all a,b\in K. If the characteristic of K is 0, then the only additive polynomials are of the form cx with c in K. But if the characteristic is p>0, then for instance, X^p and the Artin-Schreier-polynomial X^p-X are additive. I will explain the particular role that additive polynomials play in the model theory of power series fields over finite fields. This is tightly connected with the structure theory of valued function fields, which in turn also plays a crucial role for the problem of local uniformization. The latter is a local form of resolution of singularities and therefore of valuation theoretical nature. While local uniformization has been proved in characteristic 0 by Zariski in 1940, the positive characteristic case is still open, and only special cases have been solved. In all of these solutions, additive polynomials play a role. Finally, I will sketch some results on the classification of certain Artin-Schreier-extensions of valued fields and their connection with recent work of Cutkosky and Piltant on resolution of singularities in positive characteristic.

Canadian Mathematical Society Summer 2001 Meeting, Saskatoon, June 2-4, Model Theoretic Algebra Session

Let

We showed in 1989 that this axiom system is not complete. In 1998 we
developed an elementary axiom scheme which holds in maximal valued
fields of positive characteristic and describes valuation theoretic
properties of additive polynomials on such fields. This axiom scheme is
independent of the above axioms. This work has appeared in the JSL. In
1999, in joint work with L. van den Dries, we developed a particularly
simple version of this axiom scheme for **F**_{p}((t)), using
the fact that this field is locally compact. This work is to appear in
the Canad. Math. Bull.

Festkolloqium and Conference on Algebra, Model Theory and Theoretical Physics celebrating Rüdiger Göbel's 60th Birthday, University of Essen, February 2001

(joint work with Salma Kuhlmann and Saharon Shelah)

Let Gamma be a chain (= a totally ordered set), and Delta a chain with distinguished element 0

Delta

(Delta

University of Oldenburg, Germany, February 2001

Im Jahre 1965 bewies Hironaka Desingularisierung für alle algebraischen Varietä:ten über Körpern der Charakteristik 0. Das entsprechende Problem für positive Charakteristik ist bis zum heutigen Tage offen. Es wurde lediglich bis zur Dimension 3 von Abhyankar gelöst. De Jong bewies eine schwächere Aussage ("desingularization by alteration"): hier nimmt man eine endliche Erweiterung des Funktionenkörpers in Kauf.

Der ursprüngliche Ansatz von Zariski ist, zunächst einzelne Singularitäten aufzulösen, und die Lösungen dann zusammenzusetzen. Aber was bedeutet es, lokal zu desingularisieren? Dies führt direkt zum bewertungstheoretischen Begriff der "Stelle". Im Jahre 1939 bewies Zariski mit bewertungstheoretischen Methoden, dass einzelne Singularitäten in Charakteristik 0 immer aufgelöst werden können ("Local Uniformization Theorem"). Für algebraische Flächen in Charakteristik 0 konnte er dann die lokalen Lösungen zu einer globalen Desingularisierung zusammensetzen. Auch Abhyankar folgt diesem Weg, während aber Hironakas Ansatz nicht bewertungstheoretisch ist.

In den letzten Jahren hat der bewertungstheoretische Ansatz wieder an Interesse gewonnen. In meinem Vortrag werde ich zeigen, welche Rolle Hensel's Lemma in der lokalen Uniformisierung spielt, und welche neuen Ergebnisse für positive Charakteristik die Bewertungstheorie liefern kann. Ich werde das bewertungstheoretische Phänomen vorstellen, das für die Schwierigkeiten in positiver Charakteristik (in der Desingularisierung wie auch in der Modelltheorie der bewerteten Körper) verantwortlich ist. Schliesslich werde ich eine Art "bewertungstheoretisches Arbeitsprogramm" skizzieren, mit dem man das Problem der Desingularisierung in positiver Charakteristik neu in Angriff nehmen könnte.

[1] Franz-Viktor Kuhlmann: *
Valuation theoretic and model theoretic aspects of local
uniformization*, in:

Resolution
of Singularities - A Research Textbook in Tribute to Oscar
Zariski.

Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros
(eds.), Progress in Mathematics Vol.**181**, Birkhäuser
Verlag Basel (2000), 381-456

[2] Franz-Viktor Kuhlmann und Peter Roquette:

* Abhyankar places admit local uniformization in any
characteristic*, in Vorbereitung

(dvi file) ---
(postscript file)

[3] Franz-Viktor Kuhlmann: *Every place admits local uniformization in
a finite extension of the function field*, in Vorbereitung

Canadian Mathematical Society Winter Meeting, Vancouver, December 2000

In 1939, Zariski proved the Local Uniformization Theorem for places of algebraic function fields over ground fields of characteristic 0. Later, he used this theorem to prove resolution of singularities for surfaces in characteristic 0. Apart from Abhyankar's results for dimension up to 3 and de Jong's desingularization by alteration, not much has been known for positive characteristic.

We prove that every place of an algebraic function field F|K of
arbitrary characteristic admits local uniformization, provided that the
sum of the rational rank of its value group and the transcendence degree
of its residue field over K is equal to the transcendence degree of
F|K (we call such places *Abhyankar places*). Further, we show
that finitely many such places admit simultaneous local uniformization
if they have isomorphic value groups. Since Abhyankar places lie dense
in the Zariski space of all places of F|K with respect to the patch
topology, simultaneous local uniformization of any finite number of them
might open a way to pass from local uniformization to resolution of
singularities.

Further, we prove that every place of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension F' of F. This fact actually follows from de Jong's result. But we can show in addition that F'|F can be chosen to be Galois. Alternatively, F'|F can be chosen to satisfy a valuation theoretical condition which is very natural in positive characteristic. Our proofs are based solely on valuation theoretical theorems, which are of fundamental importance in positive characteristic.

We also indicate certain analogues of our results for the arithmetic case.

[1] Franz-Viktor Kuhlmann: *On local uniformization in arbitrary
characteristic*, The Fields Institute Preprint Series,
Toronto (July 1997)

(postscript file)

[2]
Franz-Viktor Kuhlmann and Peter Roquette:

* Abhyankar places admit local uniformization in any
characteristic*, in preparation

(dvi file) ---
(postscript file)

[3] Franz-Viktor Kuhlmann: *Every place admits local uniformization in
a finite extension of the function field*, in preparation

Model Theory Meeting, Oberwolfach, January 2000

[1] Kuhlmann, F.-V.: *Elementary properties of power series fields
over finite fields*, to appear in J. Symb. Logic; prepublication in:
Structures Algebriques Ordonnees, Seminaire Paris VII (1997)
(postscript file)

Talk at the conference "Model Theory of Henselian Valued Fields", Edinburgh, May 1999

I shall summarize the Ax-Kochen-Ershov principles and the relative quantifier elimination results which are known for henselian valued fields. Ax-Kochen-Ershov principles are known for the following classes:

- henselian fields of residue characteristic 0,
- henselian finitely ramified fields (includes formally p-adic fields of fixed p-rank),
- tame fields (includes algebraically closed fields and, more generally, algebraically maximal Kaplansky fields).

All tame fields are perfect. For imperfect valued fields, not many positive results are known; I shall sketch the problems one encounters when dealing with them. Similar problems appear for fields of mixed characteristic which are not finitely ramified, like certain infinite algebraic extensions of the p-adics.

Relative QE in various forms is known for all above mentioned classes of
valued fields, except for the tame fields in general. I shall describe a
form of QE which was invented by Basarab [1] and refined by myself [3].
The idea is to show QE relative to a structure naturally associated to
every valued fields, which carries more information than value group and
residue field. I called this structure ``amc-structure'' since it
encodes additive and multiplicative congruences which hold in the field.
If O and M denote valuation ring and ideal of K, then the amc-structure
(of level 0) is the residue field O/M and the group K^{x}/(1+M)
together with a natural group homomorphism (O/M)^{x} ->
K^{x}/(1+M) whose cokernel is the
value group. To cover the case of tame
fields (or imperfect fields), one would have to add something which
takes care of the structure induced by additive polynomials in the
field. At present, there is no solution known for this problem;
I shall describe the responsible stumbling block in detail.

For henselian fields of mixed characteristic, one has to use amc-structures of higher level; I shall give their definition. The class of all henselian fields of mixed characteristic admits QE relative to these structures, because they take care of the essential positive characteristic part of the fields. In special cases, one would like to get away with much less complex reducts of the amc-structures. And indeed, the Macintyre power predicates turn out to be such a reduct suitable for the case of the p-adics.

For a summary of QE for the p-adics via cell decomposition (and its use to show the rationality of the Igusa local zeta functions), I recommend [4].

[1] Basarab, S. A.: *Relative elimination of quantifiers
for Henselian valued fields*, Annals of Pure and Applied Logic
**53** (1991), 51-74

[2] Basarab, S. A. - Kuhlmann, F.-V.: *An isomorphism theorem for
henselian algebraic extensions of valued fields*, manuscripta
mathematica **77** (1992), 113-126

(postscript file)
---
[Abstract]

[3] Kuhlmann, F.-V.: *Quantifier elimination for henselian fields
relative to additive and multiplicative congruences*,
Israel Journal of Mathematics **85** (1994), 277-306

(postscript file)

[4] Pas, J.: *Some applications of uniform p-adic cell
decomposition*, Asterisque **198-200** (1991), 265-271

AMS Annual Meeting, January 1999, San Antonio, Special Session on Model Theory

Recently, it has turned out that fundamental results of valuation theory can be used to prove new theorems in the model theory of valued fields as well as about local uniformization of algebraic varieties; cf. [1]. We sketch these results and indicate how they are used to prove Ax-Kochen-Ershov principles via embedding lemmas. We present our results about perfect valued fields of positive characteristic which were obtained in this way ([2]). We apply them to gain new insight into the structure of the Zariski-Riemann manifold, which links local uniformization to desingularization ([3]). Finally, we discuss the relation between desingularization in positive characteristic and the open problem whether the power series field

[1] Kuhlmann, F.-V.: *Valuation theoretic and model theoretic
aspects of local uniformization*, to appear in the Proceedings of
the Tirol Conference on Resolution of Singularities 1997
(postscript file)

[2] Kuhlmann, F.-V.: *The model theory of tame valued fields*,
in preparation

[3] Kuhlmann, F.-V.: *On places of algebraic function fields
in arbitrary characteristic*, in preparation

[4] Kuhlmann, F.-V.: *Elementary properties of power series fields
over finite fields*, to appear in J. Symb. Logic; prepublication in:
Structures Algebriques Ordonnees, Seminaire Paris VII (1997)
(postscript file)

AMS Annual Meeting, January 1999, San Antonio, Special Session on Singularities

Using a purely valuation theoretic approach, we have proved a ``valuative version'' of de Jong's theorem. It states that every place of a function field F|K admits local uniformization on a finite normal extension of F ([2]). Our approach has the following advantages:

1) It allows us to prove ``local improvements'': for certain places, only a Galois extension is needed. In special (but important) cases, the place admits local uniformization already on F. Furthermore, instead of asking for normal extensions, one can control the extension of the value group and of the residue field associated to the extension of F ([1],[3]). This is important for certain applications, which we describe briefly. We state our results in detail and discuss the open problems and technical difficulties.

2) It connects the problem of local uniformization in positive characteristic to the open problems in the model theory of valued fields (cf. our talk in the model theory section). We sketch the fundamental valuation theoretic results which we have applied to the problem of local uniformization ([1],[2],[3]) as well as to the model theory of valued fields in positive characteristic: the theory of algebraic valuation independence, Kaplansky's theory of immediate extensions of valued fields, the generalized Grauert--Remmert stability theorem, the structure theory of valued function fields (``henselian rationality''). We indicate analogues of these results in the recent research connected with desingularization.

Finally, we give the definition of *relative local uniformization*
which works with arbitrary valued field extensions. This constitutes an
interesting new notion in valuation theory and a nice tool to prove
local uniformization results (because it is transitive). We also discuss
its connection to ramification theory.

[1] Kuhlmann, F.-V.: *On local uniformization in arbitrary
characteristic*, The Fields Institute Preprint Series,
Toronto (July 1997)
(postscript file)

[2] Kuhlmann, F.-V.: *On local uniformization in arbitrary
characteristic I*, submitted
(postscript file)

[3] Kuhlmann, F.-V.: *On local uniformization in arbitrary
characteristic II*, preprint, Saskatoon (1998)

Colloquium, Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, December 1998

A field K is called "large" if every smooth curve over K has infinitely many K-rational points, provided it has at least one. This notion was introduced by Florian Pop in an Annals paper in 1996. There he deals with problems of inverse Galois theory (a short description will be given in the talk). Among other results, Pop proves a theorem which "approximates" the Shafarevich Conjecture (which states that the absolute Galois group of the maximal cyclotomic extension of the field of rationals is profinite free).

There are several properties of fields which are equivalent to "large". Some of them are tightly connected to my own work of the early 1990's. I state these properties and give an idea of how the equivalence is proved. It turns out that there is a nice relation to properties that origin from model theoretic algebra. These express that K is "existentially closed" in suitable extensions L, that is, every elementary sentence asserting the existence of certain elements will hold in K, provided it holds in L. We show the connection of this notion with the existence of rational points and rational places.

There are many large fields. Basic examples are the algebraically closed, real closed and p-adically closed fields (and then PAC, PRC, PpC fields and fields with universal local-global principles). I explain what "existentially closed" means for the first three examples, by the Nullstellensatz framework.

Finally, I describe a new result about large fields which can be derived in two different ways, either from my results about local uniformization, or from my theory of the space of all (rational) places of an algebraic function field.

Model Theory Meeting, Oberwolfach, October 1998

Take F|K to be a function field and P a place of F which is trivial on K. If FP=K then P is called a

Theorem 1:
Assume that K admits a henselian valuation w (or, more generally,
that K is a ``large field''). Assume further that F|K admits a
rational place P.

a) If K is perfect, then K is existentially closed in F.

b) If P is weakly uniformizable, then K is existentially
closed in F.

Part b) is well-known. Which places are weakly uniformizable? Local uniformization is not known in positive characteristic, but we can give a partial answer:

Theorem 2 (K. 1997):

Rational Abhyankar places of
rank 1 and rational discrete places are weakly uniformizable.

We call a place an **Abhyankar place** if it satisfies equality in
the
Abhyankar inequality, i.e., if

where rational rank v_{P} F is the rational rank of the value group of P.

Theorem 3 (K. 1997):

If K is perfect and P is a rational place of
F|K, then there is a finite extension *F*|F and an extension of
P to *F* such that P is still a rational place of *F*|K and
(*F*|K,P) is weakly uniformizable.

Since K is existentially closed in F if it is existentially closed
in *F*, this proves part a) of Theorem 1 via part b).
For v_{P} denoting the valuation associated with P,
and v_{P}ow its composition with w, we demonstrate that a
certain kind of existential sentences which hold in (F,v_{P}ow) will
also hold in (K,w) (although in general it cannot be expected that
(K,w) be existentially closed in (F,v_{P}ow)). To
prove this by an embedding lemma, we use a result about the density of
certain ``nice'' places in the space of all rational places of F|K,
with respect to an ``existentially constructible topology''. Part b) of
Theorem 1 can be proved in a similar way, also in the case of
large fields.

*Last update: April 11, 2016*