Franz-Viktor Kuhlmann's
Abstracts of Talks
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:
What is the connection between resolution of
singularities and decidability?
Abstract:
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
Fp((t)) of formal Laurent series over the field with p
elements is decidable. Although this field looks so similar to the field
Qp of p-adic numbers and the theory of the latter has been
shown by Ax, Kochen and Ershov to be decidable, several excellent model
theorists tried in vein to solve this problem. But this is not due to a
lack of knowledge in model theory. In contrast to Qp, the
field Fp((t)) is a valued field of positive characteristic,
and we simply do not know enough about the structure of such valued
fields. In this way, model theoretical questions have stimulated new
research in a classical area of algebra: valuation theory.
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 Denef and Schoutens. They show that if resolution of
singularities in positive characteristic holds, then at least the
existential elementary theory of Fp((t)) is decidable.
Talk at the CUNY Graduate center, New York, November 2006:
Bad places and bad valuations
Abstract:
Want to deal with the model theory of valued fields in positive
characteristic? Or prove local uniformization (a local form of
resolution of singularities) in positive characteristic? Then meet your
enemy! There are bad places and valuations, and they are the main
obstruction in the above mentioned projects. We show how the defect, a
phenomenon only appearing for valuations of positive residue
characteristic, affects the structure theory of valued function fields,
which in turn is crucial for the proof of Ax-Kochen-Ershov Principles
and of local uniformization. We produce lots of defects and construct
very complicated bad valuations on very simple fields, namely rational
function fields.
But not only in positive characteristic, bad valuations on rational
function fields are interesting. Integer parts of real closed fields are
of interest for model theorists because they are precisely the models of
open induction. Mourgues and Ressayre have shown that every real closed
field admits an integer part. On the other hand, Boughattas has
constructed an ordered field that does not admit any integer part. We
construct orderings on rational function fields that do not admit
integer parts, and valuations on rational function fields that do not
admit subrings which are complements to the valuation ring. This shows
that field arithmetic structure on fields as "simple" as rational
function fields can be very complicated.
Talk in the
Model Theory and Applications to Algebra and Analysis Programme at
the Isaac Newton Institute for
Mathematical Sciences, June 2005:
Additive polynomials
Abstract:
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:
What can the theory of valued fields say about local
uniformization?
Abstract:
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:
Dense subfields and integer parts
Abstract:
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: If a valuation v has
no coarsest non-trivial coarsening and if the residue field with respect
to every coarsening is henselian, then v is henselian.
We will also discuss the existence of integer parts and of truncation
closed embeddings in power series fields. We show that Boughattas'
ordered field which admits no integer part is not henselian. Finally, we
show that there is a real closed field that has larger cardinality than
the quotient fields of all of its integer parts. It is an open question
whether there is a real closed field that is larger than the quotient
fields of all of its integer parts, but has the same cardinality.
Seminaire de Structures Algebriques Ordonnees, Univ. Paris
7, France, May 2004:
Classification of Artin-Schreier defect extensions, and
characterizations of algebraically maximal and defectless fields
Abstract:
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:
Extensions of valuations to rational function
fields
Abstract:
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:
Additive polynomials and their role in the model theory of power
series fields over finite fields and in local uniformization
Abstract:
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
Additive polynomials and the model theory of valued fields in
positive characteristic
Abstract:
Let Fp((t)) denote the field of formal Laurent series
over the field with p elements; it carries the t-adic valuation
vt. A well known open problem in model theoretic algebra is
to find a complete recursive axiom system for the elementary theory of
Fp((t)). This would yield that it is decidable. A
similar result was shown by Ax-Kochen and Ershov for the p-adics
Qp. An adaptation of their axiom system to the case of
Fp((t)) is: ``henselian defectless valued field of
characteristic p with value group a Z-group and residue field
Fp''.
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 Fp((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
Functional equations in lexicographic products
(joint work with Salma Kuhlmann and Saharon
Shelah)
Abstract:
Let Gamma be a chain (= a totally ordered set), and Delta a chain
with distinguished element 0Delta. Given a mapping s: Gamma
-> Delta, define the support of s to be the set {gamma in Gamma;
s(gamma) not= 0Delta}. The lexicographic power
DeltaGamma (in base 0Delta) is the set
{s: Gamma -> Delta ; support(s) is wellordered}
ordered lexicographically.
In DeltaGamma, we denote by 0 the sequence with
empty support. We characterize
solutions Gamma to the following functional equations:
(DeltaGamma)<=0 order isomorphic to Gamma
DeltaGamma order isomorphic to Gamma
(DeltaGamma)<0 order isomorphic to Gamma
and discuss cases (and their applications) where such solutions do not
exist.
University of Oldenburg, Germany, February 2001
Über das Problem der Desingularisierung in positiver
Charakteristik
Abstract:
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
On local uniformization in arbitrary characteristic
Abstract:
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
Additive polynomials and the
elementary properties of Fp((t))
The abstract is available as
dvi file or as
postscript file.
[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
Quantifier elimination and Ax-Kochen-Ershov principles for
henselian fields
Abstract:
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).
Tame fields are henselian valued fields K for which the ramification
field of the extension K^sep|K is algebraically closed; here,
K^sep denotes the separable-algebraic closure of K. The model theory
of tame fields has nice applications to the theory of places of
algebraic function fields in arbitrary characteristic.
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 Kx/(1+M)
together with a natural group homomorphism (O/M)x ->
Kx/(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
Local uniformization and the model theory of valued fields in
positive characteristic
Abstract:
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 Fp((t))
(which is not a perfect field!) is decidable. We point out specific
aspects of the model theoretic problem that do not appear in the local
uniformization problem, and vice versa. E.g., getting rid of tame
ramification is essential for local uniformization, but not for
embedding problems. Vice versa, local uniformization in positive
characteristic (if it exists) would have to be generalized (!) in order
to give a solution to our embedding problems. Indeed, we need a relative
version which works over non-trivially valued ground fields (where the
valuation will in general not be discrete!). We also need additional
information about the transcendence basis chosen in the uniformization.
This demonstrates that the problems from the model theory of non-perfect
valued fields lead to the formulation of very interesting
generalizations of local uniformization. However, an example originally
given to show that the model theory of Fp((t)) is a
hard problem ([4]) also sets a limit to these generalizations.
[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
A valuation theoretic approach to local uniformization
Abstract:
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
Large Fields
Abstract:
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
Rational place = existentially closed?
Abstract:
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 rational place.
We say that (F|K,P) is weakly uniformizable if there is a model
of F|K on which P is centered at a smooth point. The following is
well-known: If K is existentially closed in F, then F|K
admits a rational place which is weakly uniformizable. For the
converse, one has:
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
trdeg F|K = rational rank vP F + trdeg FP|K,
where rational rank vP 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 vP denoting the valuation associated with P,
and vPow its composition with w, we demonstrate that a
certain kind of existential sentences which hold in (F,vPow) will
also hold in (K,w) (although in general it cannot be expected that
(K,w) be existentially closed in (F,vPow)). 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: March 31, 2008