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Franz-Viktor Kuhlmann

Publications and source records attributed to Franz-Viktor Kuhlmann.

At least 37 records · Page 2Linked to original sources
arXiv2020

Eliminating Tame Ramification: generalizations of Abhyankar's Lemma

A basic version of Abhyankar's Lemma states that for two finite extensions $L$ and $F$ of a local field $K$, if $L|K$ is tamely ramified and if the ramification index of $L|K$ divides the ramification index of $F|K$, then the compositum $L.F$ is an unramified extension of $F$. In this paper, we generalize the result to valued fields with value groups of rational rank 1, and show that the latter condition is necessary. Replacing the condition on the ramification indices by the condition that the value group of $L$ be contained in that of $F$, we generalize the result further in order to give a necessary and sufficient condition for the elimination of tame ramification of an arbitrary extension $F|K$ by a suitable algebraic extension of the base field $K$. In addition, we derive more precise ramification theoretical statements and give several examples.

math.AC↗
arXiv2019

Elimination of Ramification II: Henselian Rationality

We prove in arbitrary characteristic that an immediate valued algebraic function field $F$ of transcendence degree 1 over a tame field $K$ is contained in the henselization of $K(x)$ for a suitably chosen $x\in F$. This eliminates ramification in such valued function fields. We give generalizations of this result, relaxing the assumption on $K$. Our theorems have important applications to local uniformization and to the model theory of valued fields in positive and mixed characteristic.

math.AC↗
arXiv2018

Correction and notes to the paper "A classification of Artin-Schreier defect extensions and characterizations of defectless fields"

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

math.AC↗
arXiv2018

Construction of ball spaces and the notion of continuity

Spherically complete ball spaces provide a framework for the proof of generic fixed point theorems. For the purpose of their application it is important to have methods for the construction of new spherically complete ball spaces from given ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.

math.GN↗
arXiv2018

Chain intersection closures

We study spherical completeness of ball spaces and its stability under expansions. We introduce the notion of an ultra-diameter, mimicking diameters in ultrametric spaces. We prove some positive results on preservation of spherical completeness involving ultra-diameters with values in narrow partially ordered sets. Finally, we show that in general, chain intersection closures of ultrametric spaces with partially ordered value sets do not preserve spherical completeness.

math.LO↗
arXiv2018

On valuation independence and defectless extensions of valued fields

In this article we further develop the theory of valuation independence and study its relation with classical notions in valuation theory such as immediate and defectless extensions. We use this general theory to settle two open questions regarding vector space defectless extensions of valued fields. Additionally, we provide a characterization of such extensions within various classes of valued fields, extending results of Françoise Delon.

math.AC↗
arXiv2018

Subfields of algebraically maximal Kaplansky fields

Using the ramification theory of tame and Kaplansky fields, we show that maximal Kaplansky fields contain maximal immediate extensions of each of their subfields. Likewise, algebraically maximal Kaplansky fields contain maximal immediate algebraic extensions of each of their subfields. This study is inspired by problems that appear in henselian valued fields of rank higher than 1 when a Hensel root of a polynomial is approximated by the elements generated by a (transfinite) Newton algorithm.

math.AC↗
arXiv2017

Counting the number of distinct distances of elements in valued field extensions

The defect of valued field extensions is a major obstacle in open problems in resolution of singularities and in the model theory of valued fields, whenever positive characteristic is involved. We continue the detailed study of defect extensions through the tool of distances, which measure how well an element in an immediate extension can be approximated by elements from the base field. We show that in several situations the number of essentially distinct distances in fixed extensions, or even just over a fixed base field, is finite, and we compute upper bounds. We apply this to the special case of valued functions fields over perfect base fields. This provides important information used in forthcoming research on relative resolution problems.

math.AC↗
arXiv2016

Valuation theory of exponential Hardy fields II: Principal parts of germs in the Hardy field of o-minimal exponential expansions of the reals

We present a general structure theorem for the Hardy field of an o-minimal expansion of the reals by restricted analytic functions and an unrestricted exponential. We proceed to analyze its residue fields with respect to arbitrary convex valuations, and deduce a power series expansion of exponential germs. We apply these results to cast "Hardy's conjecture" (see \cite[p.111]{[KS]}) in a more general framework. This paper is a follow up to \cite{[K-K2]} and is partially based on unpublished results of \cite{[K-K]}. A previous version \cite{[K-K1]} (which was dedicated to Murray A. Marshall on his 60th birthday) remained unpublished. In \cite{[W]} our structure theorem for the residue fields was rediscovered and applied to the diophantine context. Due to this revived interest, we decided to rework the preprint \cite{[K-K1]} and submit it to the Proceedings Volume.

math.LO↗
arXiv2016

Notes on extremal and tame valued fields

We extend the characterization of extremal valued fields given in \cite{[AKP]} to the missing case of valued fields of mixed characteristic with perfect residue field. This leads to a complete characterization of the tame valued fields that are extremal. The key to the proof is a model theoretic result about tame valued fields in mixed characteristic. Further, we prove that in an extremal valued field of finite $p$-degree, the images of all additive polynomials have the optimal approximation property. This fact can be used to improve the axiom system that is suggested in \cite{[K1]} for the elementary theory of Laurent series fields over finite fields. Finally we give examples that demonstrate the problems we are facing when we try to characterize the extremal valued fields with imperfect residue fields. To this end, we describe several ways of constructing extremal valued fields; in particular, we show that in every $\aleph_1$ saturated valued field the valuation is a composition of extremal valuations of rank 1.

math.LO↗
arXiv2015

The model theory of separably tame valued fields

A henselian valued field $K$ is called separably tame if its separable-algebraic closure $K^{\operatorname{sep}}$ is a tame extension, that is, the ramification field of the normal extension $K^{\operatorname{sep}}|K$ is separable-algebraically closed. Every separable-algebraically maximal Kaplansky field is a separably tame field, but not conversely. In this paper, we prove Ax-Kochen-Ershov Principles for separably tame fields. This leads to model completeness and completeness results relative to the value group and residue field. As the maximal immediate extensions of separably tame fields are in general not unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax-Kochen-Ershov Principles. Our approach also yields alternate proofs of known results for separably closed valued fields.

math.LO↗
arXiv2014

The algebra and model theory of tame valued fields

A henselian valued field $K$ is called a tame field if its algebraic closure $\tilde{K}$ is a tame extension, that is, the ramification field of the normal extension $\tilde{K}|K$ is algebraically closed. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We develop the algebraic theory of tame fields and then prove Ax--Kochen--Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax--Kochen--Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.

math.AC↗
arXiv2013

Henselian Elements

Henselian elements are roots of polynomials which satisfy the conditions of Hensel's Lemma. In this paper we prove that for a finite field extension $(F|L,v)$, if $F$ is contained in the absolute inertia field of $L$, then the valuation ring $\mathcal O_F$ of $(F,v)$ is generated as an $\mathcal O_L$-algebra by henselian elements. Moreover, we give a list of equivalent conditions under which $\mathcal O_F$ is generated over $\mathcal O_L$ by finitely many henselian elements. We prove that if the chain of prime ideals of $\mathcal O_L$ is well-ordered, then these conditions are satisfied. We give an example of a finite valued inertial extension $(F|L,v)$ for which $\mathcal O_F$ is not a finitely generated $\mathcal O_L$-algebra. We also present a theorem that relates the problem of local uniformization with the theory of henselian elements.

math.AC↗
arXiv2013

Symmetrically complete ordered sets, abelian groups and fields

We characterize and construct linearly ordered sets, abelian groups and fields that are {\emph symmetrically complete}, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach's Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. We show how to extend any given ordered set, abelian group or field to one that is symmetrically complete. A main part of the paper establishes a detailed study of the cofinalities in cuts.

math.LO↗