77 results on '"Franz-Viktor Kuhlmann"'
Search Results
2. Tame Key polynomials
- Author
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Arpan Dutta and Franz-Viktor Kuhlmann
- Subjects
Algebra and Number Theory ,FOS: Mathematics ,12J20, 13A18, 12J25 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra - Abstract
We introduce a new method of constructing complete sequences of key polynomials for simple extensions of tame fields. In our approach the key polynomials are taken to be the minimal polynomials over the base field of suitably constructed elements in its algebraic closure, with the extensions generated by them forming an increasing chain. In the case of algebraic extensions, we generalize the results to countably generated infinite tame extensions over henselian but not necessarily tame fields. In the case of transcendental extensions, we demonstrate the central role that is played by the implicit constant fields, which reveals the tight connection with the algebraic case.
- Published
- 2022
3. Elimination of ramification II: Henselian rationality
- Author
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Franz-Viktor Kuhlmann
- Subjects
Model theory ,Algebraic function field ,Pure mathematics ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,12J10 ,Rationality ,Field (mathematics) ,0102 computer and information sciences ,Function (mathematics) ,Transcendence degree ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,010201 computation theory & mathematics ,FOS: Mathematics ,0101 mathematics ,Uniformization (set theory) ,Mathematics - Abstract
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., new improved version
- Published
- 2019
4. Valued fields with finitely many defect extensions of prime degree
- Author
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Franz-Viktor Kuhlmann
- Subjects
Pure mathematics ,Algebra and Number Theory ,Property (philosophy) ,Group (mathematics) ,Applied Mathematics ,Open problem ,Prime degree ,Field (mathematics) ,Mathematics - Logic ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,12J10, 13A18 ,Residue field ,Hull ,FOS: Mathematics ,Logic (math.LO) ,Value (mathematics) ,Mathematics - Abstract
We prove that a valued field of positive characteristic [Formula: see text] that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP2 fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has [Formula: see text]-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin–Schreier defect extensions.
- Published
- 2021
5. Elementary Properties of Power Series Fields over Finite Fields.
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Franz-Viktor Kuhlmann
- Published
- 2001
- Full Text
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6. Density of Composite Places in Function Fields and Applications to Real Holomorphy Rings
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Eberhard Becker, Franz‐Viktor Kuhlmann, and Katarzyna Kuhlmann
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General Mathematics ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Primary 12J10, 12J15, secondary 12D15, 12J25 - Abstract
Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we prove that the places that are composite with extensions of $\wp$ to finite extensions of $K$ lie dense in the space of all places of $F$, in a strong sense. We apply the result to the case of $K=R$ any real closed field and the fixed place on $R$ being its natural (finest) real place. This leads to a new description of the real holomorphy ring of $F$ which can be seen as an analogue to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space $M(F)$ of all $\R$-places of $F$ (places with residue field contained in $\R$), its subspace of all $\R$-places of $F$ that are composite with the natural $\R$-place of $R$, and the topological space of all $R$-rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space $M(F)$ has only finitely many topological components., 29 pages
- Published
- 2020
7. Correction and notes to the paper 'A classification of Artin–Schreier defect extensions and characterizations of defectless fields'
- Author
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Franz-Viktor Kuhlmann
- Subjects
Discrete mathematics ,Lemma (mathematics) ,14B05 ,General Mathematics ,13A18 ,010102 general mathematics ,12J10 ,Mistake ,Commutative Algebra (math.AC) ,Linearly disjoint ,Mathematics - Commutative Algebra ,01 natural sciences ,Primary 12J10, 13A18, Secondary 12J25, 12L12, 14B05 ,Field extension ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,12J25 ,12L12 ,Mathematics - Abstract
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.
- Published
- 2019
8. Fixed point theorems for spaces with a transitive relation
- Author
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Franz-Viktor Kuhlmann and Katarzyna Kuhlmann
- Subjects
Discrete mathematics ,Transitive relation ,Applied Mathematics ,010102 general mathematics ,Fixed-point theorem ,Prewellordering ,Transitive set ,Fixed-point property ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Critical point (set theory) ,0101 mathematics ,Analysis ,Mathematics - Published
- 2017
9. Coincidence point theorems for ball spaces and their applications
- Author
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Franz-Viktor Kuhlmann, Katarzyna Kuhlmann, and Fatma Sonaallah
- Published
- 2017
10. A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures
- Author
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Hanna Ćmiel, Franz-Viktor Kuhlmann, and Katarzyna Kuhlmann
- Subjects
Pure mathematics ,Algebra and Number Theory ,Applied Mathematics ,Least-upper-bound property ,54A05, 54H25 (Primary), 03E75, 06A05, 06A06, 06B23, 06B99, 06F20, 12J15, 12J20, 13A18, 47H09, 47H10, 54C10, 54C60, 54E50 (Secondary) ,General Topology (math.GN) ,Topological space ,Space (mathematics) ,Computational Mathematics ,Metric space ,Tychonoff's theorem ,Compact space ,Completeness (order theory) ,FOS: Mathematics ,Geometry and Topology ,Ultrametric space ,Analysis ,Mathematics - General Topology ,Mathematics - Abstract
With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster--Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.
- Published
- 2019
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11. Eliminating Tame Ramification: generalizations of Abhyankar's Lemma
- Author
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Arpan Dutta and Franz-Viktor Kuhlmann
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Lemma (mathematics) ,Pure mathematics ,General Mathematics ,Mathematics::Number Theory ,010102 general mathematics ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,12J20, 13A18, 12J25 ,0101 mathematics ,Mathematics - Abstract
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.
- Published
- 2019
- Full Text
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12. Valuations on rational function fields that are invariant under permutation of the variables
- Author
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Katarzyna Kuhlmann, Franz-Viktor Kuhlmann, and C. Vişan
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Discrete mathematics ,Computer Science::Computer Science and Game Theory ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,010102 general mathematics ,Rational function ,01 natural sciences ,0103 physical sciences ,Quantitative Biology::Populations and Evolution ,010307 mathematical physics ,0101 mathematics ,Invariant (mathematics) ,Discrete valuation ,Finite set ,Mathematics ,Valuation (finance) - Abstract
We study and characterize the class of valuations on rational functions fields that are invariant under permutation of the variables and can be extended to valuations with the same property whenever a finite number of new variables is adjoined. The Gaus valuation is in this class, which constitutes a natural generalization of the concept of Gaus valuation. Further, we apply our characterization to show that the most common ad hoc generalization of the Gaus valuation is also in this class.
- Published
- 2016
13. SEPARABLY CLOSED VALUED FIELDS: QUANTIFIER ELIMINATION
- Author
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Franz-Viktor Kuhlmann and Sylvy Anscombe
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Pure mathematics ,Rank (linear algebra) ,Logic ,Approximation property ,Laurent series ,010102 general mathematics ,Field (mathematics) ,16. Peace & justice ,01 natural sciences ,Philosophy ,Finite field ,Residue field ,0103 physical sciences ,Quantifier elimination ,Elementary theory ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
It is proved in this article that the theory of separably closed nontrivially valued fields of characteristic p > 0 and imperfection degree e > 0 (e ≤ ∞) has quantifier elimination in the language ${{\cal L}_{p,{\rm{div}}}} = \{ + , - , \times ,0,1\} \cup {\{ {\lambda _{n,j}}(x;{y_1}, \ldots ,{y_n})\} _{0 \le n < \omega ,0 \le j < {p^n}}} \cup \{ |\}$; in particular, when e is finite, the corresponding theory has quantifier elimination in the language ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.
- Published
- 2016
14. On maximal immediate extensions of valued fields
- Author
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Franz-Viktor Kuhlmann and Anna Blaszczok
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Discrete mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Published
- 2016
15. The valuation theory of deeply ramified fields and its connection with defect extensions
- Author
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Franz-Viktor Kuhlmann and Anna Rzepka
- Subjects
Applied Mathematics ,General Mathematics ,FOS: Mathematics ,12J10, 12J25 ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra - Abstract
We study in detail the valuation theory of deeply ramified fields and introduce and investigate several other related classes of valued fields. Further, a classification of defect extensions of prime degree of valued fields that was earlier given only for the equicharacteristic case is generalized to the case of mixed characteristic by a unified definition that works simultaneously for both cases. It is shown that deeply ramified fields and the other valued fields we introduce only admit one of the two types of defect extensions, namely the ones that appear to be more harmless in open problems such as local uniformization and the model theory of valued fields in positive characteristic. We use our knowledge about such defect extensions to give a new, valuation theoretic proof of the fact that algebraic extensions of deeply ramified fields are again deeply ramified. We also prove finite descent, and under certain conditions even infinite descent, for deeply ramified fields. These results are also proved for two other related classes of valued fields. The classes of valued fields under consideration can be seen as generalizations of the class of tame valued fields. Our paper supports the hope that it will be possible to generalize to deeply ramified fields several important results that have been proven for tame fields and were at the core of partial solutions of the two open problems mentioned above., Paper is to appear in Transactions AMS. Anna's family name has changed from "Blaszczok" to "Rzepka". Parts of the original version were taken out and are to be published in later papers. In turn, this version contains significant new theorems
- Published
- 2018
16. Chain intersection closures
- Author
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Wiesław Kubiś and Franz-Viktor Kuhlmann
- Subjects
Pure mathematics ,010102 general mathematics ,General Topology (math.GN) ,Mathematics - Logic ,01 natural sciences ,010101 applied mathematics ,06A06, 54A05 ,FOS: Mathematics ,Geometry and Topology ,Ball (mathematics) ,0101 mathematics ,Logic (math.LO) ,Partially ordered set ,Ultrametric space ,Mathematics ,Mathematics - General Topology - Abstract
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., 11 pages
- Published
- 2018
17. The Caristi–Kirk Fixed Point Theorem from the point of view of ball spaces
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Matthias Paulsen, Franz-Viktor Kuhlmann, and Katarzyna Kuhlmann
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Pure mathematics ,Quantitative Biology::Neurons and Cognition ,Applied Mathematics ,010102 general mathematics ,Fixed-point theorem ,Mathematics::Geometric Topology ,01 natural sciences ,010101 applied mathematics ,Metric space ,If and only if ,Modeling and Simulation ,Geometry and Topology ,Ball (mathematics) ,0101 mathematics ,Mathematics - Abstract
We take a fresh look at the important Caristi-Kirk Fixed Point Theorem and link it to the recently developed theory of ball spaces, which provides generic fixed point theorems for contracting functions in a number of applications including, but not limited to, metric spaces. The connection becomes clear from a proof of the Caristi-Kirk Theorem given by J.-P. Penot in 1976. We define Caristi-Kirk ball spaces and use a generic fixed point theorem to reprove the Caristi-Kirk Theorem. Further, we show that a metric space is complete if and only if all of its Caristi-Kirk ball spaces are spherically complete.
- Published
- 2018
18. The model theory of separably tame valued fields
- Author
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Koushik Pal and Franz-Viktor Kuhlmann
- Subjects
Model theory ,Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,010102 general mathematics ,Closure (topology) ,Normal extension ,Field (mathematics) ,Mathematics - Logic ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Residue field ,Completeness (order theory) ,Primary 03C10, 12J10, Secondary 03C60, 12J20 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) ,Mathematics ,Valuation (algebra) - Abstract
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., 30 pages. arXiv admin note: substantial text overlap with arXiv:1304.0194
- Published
- 2016
19. Selected methods for the classification of cuts, and their applications
- Author
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Franz-Viktor Kuhlmann
- Subjects
Pure mathematics ,Group (mathematics) ,Mathematics - Rings and Algebras ,Primary 06F20, 12J15, secondary 13J30, 20F60 ,Cofinality ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Valuation ring ,Ordered field ,Rings and Algebras (math.RA) ,FOS: Mathematics ,General Earth and Planetary Sciences ,Ball (mathematics) ,Abelian group ,General Environmental Science ,Mathematics - Abstract
We consider four approaches to the analysis of cuts in ordered abelian groups and ordered fields, their interconnection, and various applications. The notions we discuss are: ball cuts, invariance group, invariance valuation ring, and cut cofinality.
- Published
- 2018
- Full Text
- View/download PDF
20. Construction of ball spaces and the notion of continuity
- Author
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René Bartsch, Franz-Viktor Kuhlmann, and Katarzyna Kuhlmann
- Subjects
Pure mathematics ,Algebra and Number Theory ,Computer science ,Applied Mathematics ,Work (physics) ,Coproduct ,General Topology (math.GN) ,Fixed-point theorem ,Topological category ,Set (abstract data type) ,Simple (abstract algebra) ,TheoryofComputation_ANALYSISOFALGORITHMSANDPROBLEMCOMPLEXITY ,Ball (bearing) ,FOS: Mathematics ,Geometry and Topology ,Analysis ,Quotient ,Primary 54A05, Secondary 54H25, 12J15, 03E20, 18B05 ,Mathematics - General Topology ,ComputingMethodologies_COMPUTERGRAPHICS - Abstract
Spherically complete ball spaces provide a simple framework for the encoding of completeness properties of various spaces and ordered structures. This allows to prove generic versions of theorems that work with these completeness properties, such as fixed point theorems and related results. For the purpose of applying the generic theorems, it is important to have methods for the construction of new spherically complete ball spaces from existing 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.
- Published
- 2018
- Full Text
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21. Ordered Algebraic Structures and Related Topics
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Franz-Viktor Kuhlmann and Salma Kuhlmann
- Subjects
Power series ,Conjecture ,Diophantine equation ,010102 general mathematics ,Context (language use) ,0102 computer and information sciences ,01 natural sciences ,Exponential function ,Combinatorics ,010201 computation theory & mathematics ,Hardy field ,0101 mathematics ,Structured program theorem ,Mathematics ,Analytic function - Abstract
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.
- Published
- 2017
22. Corrections and notes to 'Value groups, residue fields and bad places of rational function fields'
- Author
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Anna Blaszczok and Franz-Viktor Kuhlmann
- Subjects
Residue (complex analysis) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Calculus ,Rational function ,Mathematics - Abstract
We correct mistakes in a 2004 paper by the second author and report on recent new developments which settle cases left open in that paper.
- Published
- 2015
23. Counting the number of distinct distances of elements in valued field extensions
- Author
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Franz-Viktor Kuhlmann and Anna Blaszczok
- Subjects
Model theory ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Resolution of singularities ,Field (mathematics) ,Function (mathematics) ,Base (topology) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Measure (mathematics) ,Field extension ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Element (category theory) ,12J10, 12J25 ,Mathematics - Abstract
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. In particular, this provides important information used in forthcoming research on the ramification theory of two-dimensional valued function fields.
- Published
- 2017
24. Dense subfields of henselian fields, and integer parts
- Author
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Franz-Viktor Kuhlmann
- Published
- 2017
25. The algebra and model theory of tame valued fields
- Author
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Franz-Viktor Kuhlmann
- Subjects
Algebraic function field ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Normal extension ,Field (mathematics) ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Algebraic closure ,Algebra ,Residue field ,Algebraic theory ,0103 physical sciences ,FOS: Mathematics ,12J10, 12J15 ,010307 mathematical physics ,0101 mathematics ,Algebraically closed field ,Valuation (algebra) ,Mathematics - Abstract
A henselian valued field K is called a tame field if its algebraic closure K ~ ${\tilde{K}}$ is a tame extension, that is, the ramification field of the normal extension K ~ | K ${\tilde{K}|K}$ 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.
- Published
- 2014
26. Valuation Theory and Its Applications
- Author
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Zoé Chatzidakis, Franz-Viktor Kuhlmann, Jochen Koenigsmann, and Florian Pop
- Subjects
General Medicine - Published
- 2014
27. A common generalization of metric, ultrametric and topological fixed point theorems
- Author
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Franz-Viktor Kuhlmann and Katarzyna Kuhlmann
- Subjects
Discrete mathematics ,Metric space ,Generalization ,Applied Mathematics ,General Mathematics ,Metric (mathematics) ,Fixed-point theorem ,Topology ,Ultrametric space ,Mathematics ,Ordered field - Abstract
We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's fixed point theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not involving any metrics. We demonstrate its applications to the metric, ultrametric and topological cases, and to ordered abelian groups and fields.
- Published
- 2012
28. Characterization of extremal valued fields
- Author
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Salih Azgin, Florian Pop, and Franz-Viktor Kuhlmann
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Polynomial ,Pure mathematics ,Mathematics::Commutative Algebra ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Zero (complex analysis) ,Characterization (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Valuation ring ,Image (mathematics) ,010104 statistics & probability ,primary: 12J10, secondary: 12E30 ,FOS: Mathematics ,0101 mathematics ,Element (category theory) ,Value (mathematics) ,Mathematics - Abstract
We characterize those valued fields for which the image of the valuation ring under every polynomial in several variables contains an element of maximal value, or zero., Comment: 12 pages
- Published
- 2012
29. Metrizability of Spaces of ℝ-places of Function Fields of Transcendence Degree 1 Over Real Closed Fields
- Author
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Michał Machura, Franz-Viktor Kuhlmann, and Katarzyna Osiak
- Subjects
Discrete mathematics ,Real closed field ,Pure mathematics ,Algebra and Number Theory ,Countable set ,Field (mathematics) ,Function (mathematics) ,Transcendence degree ,Rational function ,Space (mathematics) ,Function field ,Mathematics - Abstract
In this article we discuss the following question “When do different orderings of the rational function field R(X) (where R is a real closed field) induce the same ℝ-place?” We use this to show that if R contains a dense real closed subfield R′, then the spaces of ℝ-places of R(X) and R′(X) are homeomorphic. For the function field K = R(X), we prove that its space M(K) of ℝ-places is metrizible if and only if R contains a countable dense subfield. Moreover, we show that this condition is neccessary for the metrizability of M(F) for any function field F of transcendence degree 1 over R.
- Published
- 2011
30. Maps on Ultrametric Spaces, Hensel's Lemma, and Differential Equations Over Valued Fields
- Author
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Franz-Viktor Kuhlmann
- Subjects
Discrete mathematics ,Lemma (mathematics) ,Algebra and Number Theory ,Aubin–Lions lemma ,Céa's lemma ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Implicit function theorem ,Surjective function ,12J10, 12J20 ,FOS: Mathematics ,Differential algebra ,Ultrametric space ,Hensel's lemma ,Mathematics - Abstract
We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic., 47 pages
- Published
- 2011
31. Towers of complements to valuation rings and truncation closed embeddings of valued fields
- Author
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Antongiulio Fornasiero, Salma Kuhlmann, and Franz-Viktor Kuhlmann
- Subjects
Power series ,Complement to valuation ring ,Truncation ,Completion of an ordered group ,Fields of generalized power series ,Field (mathematics) ,0102 computer and information sciences ,Commutative Algebra (math.AC) ,01 natural sciences ,Valuation ring ,Valued field ,Combinatorics ,Residue field ,FOS: Mathematics ,0101 mathematics ,Mathematics ,Valuation (algebra) ,Algebra and Number Theory ,Mathematics::Commutative Algebra ,Group (mathematics) ,010102 general mathematics ,Mathematics - Logic ,Truncation closed embedding ,Mathematics - Commutative Algebra ,Primary: 12J10, 12J15, 12L12, 13A18, Secondary: 03C60, 12F05, 12F10, 12F20 ,010201 computation theory & mathematics ,Embedding ,Logic (math.LO) - Abstract
We study necessary and sufficient conditions for a valued field K with value group G and residue field k (with char K = char k ) to admit a truncation closed embedding in the field of generalized power series k ( ( G , f ) ) (with factor set f). We show that this is equivalent to the existence of a family (tower of complements) of k-subspaces of K which are complements of the (possibly fractional) ideals of the valuation ring, and satisfying certain natural conditions. If K is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.
- Published
- 2010
- Full Text
- View/download PDF
32. Every place admits local uniformization in a finite extension of the function field
- Author
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Franz-Viktor Kuhlmann and Hagen Knaf
- Subjects
Algebraic function field ,Discrete mathematics ,Mathematics(all) ,General Mathematics ,Purely inseparable extension ,Normal extension ,Abelian extension ,Separable extension ,Algebraic number field ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Primary extension ,Mathematics - Algebraic Geometry ,12J10, 14E15 ,FOS: Mathematics ,Galois extension ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We prove that every place P of an algebraic function field F|K of arbitrary characteristic admits local uniformization in a finite extension E of F. We show that E|F can be chosen to be Galois, after a finite purely inseparable extension of the ground field K. Instead of being Galois, the extension can also be chosen such that the induced extension EP|FP of the residue fields is purely inseparable and the value group of F only gets divided by the residue characteristic. If F lies in the completion of an Abhyankar place, then no extension of F is needed. Our proofs are based solely on valuation theoretical theorems, which are of particular importance in positive characteristic. They are also applicable when working over a subring R of K and yield similar results if R is regular and of dimension smaller than 3., Comment: 27 pages
- Published
- 2009
33. Abhyankar places admit local uniformization in any characteristic
- Author
-
Franz-Viktor Kuhlmann and Hagen Knaf
- Subjects
Algebraic function field ,Discrete mathematics ,General Mathematics ,Transcendence degree ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,Nagata ring ,Separable space ,Combinatorics ,Mathematics - Algebraic Geometry ,12J10, 14E15 ,Residue field ,FOS: Mathematics ,Krull dimension ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
We prove that every place $P$ 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 $FP$ over $K$ is equal to the transcendence degree of $F|K$, and the extension $FP|K$ is separable. We generalize this result to the case where $P$ dominates a regular local Nagata ring $R\subseteq K$ of Krull dimension $\dim R\leq 2$, assuming that the valued field $(K,v_P)$ is defectless, the factor group $v_P F/v_P K$ is torsion-free and the extension of residue fields $FP|KP$ is separable. The results also include a form of monomialization. Further, we show that in both cases, finitely many Abhyankar places admit simultaneous local uniformization on an affine scheme if they have value groups isomorphic over $v_P K$., 21 pages, submitted
- Published
- 2005
34. Formal power series with cyclically ordered exponents
- Author
-
M. Giraudet, Franz-Viktor Kuhlmann, and G. Leloup
- Subjects
Discrete mathematics ,Monomial ,Mathematics::Commutative Algebra ,Formal power series ,General Mathematics ,Formal group ,Predicate (mathematical logic) ,Quotient ,Mathematics ,Valuation function - Abstract
We define and study a notion of ring of formal power series with exponents in a cyclically ordered group. Such a ring is a quotient of various subrings of classical formal power series rings. It carries a two variable valuation function. In the particular case where the cyclically ordered group is actually totally ordered, our notion of formal power series is equivalent to the classical one in a language enriched with a predicate interpreted by the set of all monomials.
- Published
- 2005
35. Value groups, residue fields, and bad places of rational function fields
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Power series ,Residue (complex analysis) ,Pure mathematics ,Tensor product of fields ,Applied Mathematics ,General Mathematics ,Rational function ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Algebraic closure ,Ground field ,Algebra ,Residue field ,FOS: Mathematics ,Finitely-generated abelian group ,12J10 (Primary), 12J15 (Secondary), 16W60 ,Mathematics - 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 several constructions of extensions whose corresponding value group and residue field extensions are not finitely generated. 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. Finally, we apply our methods to power series fields and the $p$-adics., Comment: 41 pages
- Published
- 2004
36. Henselian Elements
- Author
-
Franz-Viktor Kuhlmann and Josnei Novacoski
- Subjects
Algebra and Number Theory ,Mathematics::Commutative Algebra ,Mathematics::Number Theory ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra - Abstract
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.
- Published
- 2013
37. Dynamic computations inside the algebraic closure of a valued field
- Author
-
Franz-Viktor Kuhlmann, Henri Lombardi, and Hervé Perdry
- Published
- 2003
38. Symmetrically complete ordered sets, abelian groups and fields
- Author
-
Katarzyna Kuhlmann, Franz-Viktor Kuhlmann, and Saharon Shelah
- Subjects
Power series ,General Mathematics ,Modulo ,010102 general mathematics ,Fixed-point theorem ,Elementary abelian group ,Mathematics - Logic ,01 natural sciences ,Primary 12J15, 06F20, 03E04, Secondary 06A05 ,Rank of an abelian group ,010101 applied mathematics ,Combinatorics ,Intersection ,Bounded function ,FOS: Mathematics ,0101 mathematics ,Abelian group ,Logic (math.LO) ,Mathematics - Abstract
We characterize linearly ordered sets, abelian groups and fields that are 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. This gives us a direct route to the construction of symmetrically complete ordered abelian groups and fields, modulo an analogous construction at the level of ordered sets; in particular, this gives an alternative approach to the construction of symmetrically complete fields in [12].
- Published
- 2013
39. Functorial equations for lexicographic products
- Author
-
Salma Kuhlmann, Franz-Viktor Kuhlmann, and Saharon Shelah
- Subjects
Discrete mathematics ,Pure mathematics ,Functor ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Regular polygon ,0102 computer and information sciences ,Lexicographical order ,01 natural sciences ,Lexicography ,Chain (algebraic topology) ,010201 computation theory & mathematics ,Order (group theory) ,ddc:510 ,0101 mathematics ,Mathematics - Abstract
We generalize the main result of [K–K–S] concerning the convex embeddings of a chain Γ in a lexicographic power Δ Γ . For a fixed non-empty chain Δ, we derive necessary and sufficient conditions for the existence of non-empty solutions Γ to each of the lexicographic functorial equations (Δ Γ ) ≤0 � Γ , (Δ Γ ) � Γa nd (Δ Γ )
- Published
- 2003
40. Images of Additive Polynomials in 𝔽q((t)) Have the Optimal Approximation Property
- Author
-
Franz Viktor Kuhlmann and Lou van den Dries
- Subjects
Equioscillation theorem ,Pure mathematics ,Approximation property ,General Mathematics ,Mathematics - Abstract
We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.
- Published
- 2002
41. Construction du hensélisé d'un corps valué
- Author
-
Franz-Viktor Kuhlmann and Henri Lombardi
- Subjects
Mots Clés: corps valué ,Algebra and Number Theory ,Mathematics - Commutative Algebra ,12J10, 12F05, 13J15, 12Y05, 03F65 ,Humanities ,mathématiques constructives ,Hensélisation ,Mathematics - Abstract
We give an explicit construction of the henselization of a valued field, with a constructive proof. It is analogous to the construction of the real closure of a discrete ordered field. Nous donnons une construction explicite, et constructivement prouv\'ee, du hens\'elis\'e d'un corps valu\'e. Cette construction peut \^etre consid\'er\'ee comme l'analogue, dans le cas valu\'e, de la construction de la cl\^oture r\'e\'eelle d'un corps ordonn\'e., Comment: in French
- Published
- 2000
42. Algebraic independence of elements in immediate extensions of valued fields
- Author
-
Franz-Viktor Kuhlmann and Anna Blaszczok
- Subjects
Pure mathematics ,Algebra and Number Theory ,Group (mathematics) ,Field (mathematics) ,Rational function ,Transcendence degree ,Commutative Algebra (math.AC) ,16. Peace & justice ,Cofinality ,Mathematics - Commutative Algebra ,FOS: Mathematics ,Countable set ,Algebraic independence ,Algebraic number ,12J10, 12J25 ,Mathematics - Abstract
Refining a constructive combinatorial method due to MacLane and Schilling, we give several criteria for a valued field that guarantee that all of its maximal immediate extensions have infinite transcendence degree. If the value group of the field has countable cofinality, then these criteria give the same information for the completions of the field. The criteria have applications to the classification of valuations on rational function fields. We also apply the criteria to the question of which extensions of a maximal valued field, algebraic or of finite transcendence degree, are again maximal. In the case of valued fields of infinite p-degree, we obtain the worst possible examples of nonuniqueness of maximal immediate extensions: fields which admit an algebraic maximal immediate extension as well as one of infinite transcendence degree.
- Published
- 2013
43. Embedding theorems for spaces of $\R$-places of rational function fields and their products
- Author
-
Franz-Viktor Kuhlmann and Katarzyna Kuhlmann
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,FOS: Mathematics ,Embedding ,010103 numerical & computational mathematics ,Rational function ,Primary 12J15, Secondary 12J25 ,0101 mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Mathematics - Abstract
We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\R$-places.
- Published
- 2013
44. The relative approximation degree in valued function fields
- Author
-
Franz-Viktor Kuhlmann and Izabela Vlahu
- Subjects
Degree (graph theory) ,Mathematics::Commutative Algebra ,General Mathematics ,Modulo ,010102 general mathematics ,Function (mathematics) ,Transcendence degree ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Uniformization (probability theory) ,Algebra ,Field extension ,Primary 12J10, Secondary 12J20 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Element (category theory) ,Function field ,Mathematics - Abstract
We continue the work of Kaplansky on immediate valued field extensions and determine special properties of elements in such extensions. In particular, we are interested in the question when an immediate valued function field of transcendence degree 1 is henselian rational (i.e., generated, modulo henselization, by one element). If so, then wild ramification can be eliminated in this valued function field. The results presented in this paper are crucial for the first author's proof of henselian rationality over tame fields, which in turn is used in his work on local uniformization.
- Published
- 2013
45. Valuation theory of exponential Hardy fields II: Principal parts of germs in the Hardy field of o-minimal exponential expansions of the reals
- Author
-
Franz-Viktor Kuhlmann and Salma Kuhlmann
- Subjects
FOS: Mathematics ,Mathematics - Logic ,Logic (math.LO) - Abstract
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., 15 pages, to appear in AMS Contemporary Mathematics (CONM); Proceedings of the Conference on Ordered Algebraic Structures and Related Topics. Edited by: F. Broglia, F. Delon, M. Dickmann, D. Gondard, and V. Powers (2016)
- Published
- 2012
46. Correction to A common generalization of metric, ultrametric and topological fixed point theorems [Forum Math. 27 (2015), 303–327]
- Author
-
Franz-Viktor Kuhlmann and Katarzyna Kuhlmann
- Subjects
Discrete mathematics ,Generalization ,Applied Mathematics ,General Mathematics ,Metric (mathematics) ,Fixed-point theorem ,Topology ,Ultrametric space ,Mathematics - Abstract
Corrections to http://dx.doi.org/10.1515/forum-2012-0060.
- Published
- 2013
47. A correction to Epp?s paper ?Elimination of wild ramification?
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Pure mathematics ,Mathematics::Commutative Algebra ,General Mathematics ,Ramification (botany) ,FOS: Mathematics ,12J10 ,Discrete valuation ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,Statistics::Computation ,Mathematics - Abstract
We fill a gap in the proof of one of the central theorems in Epp's paper, concerning $p$-cyclic extensions of complete discrete valuation rings., Comment: 3 pages
- Published
- 2003
48. The defect
- Author
-
Franz-Viktor Kuhlmann
- Published
- 2010
49. Valuation theoretic and model theoretic aspects of local uniformization
- Author
-
Franz-Viktor Kuhlmann
- Subjects
12J10, 03C60, 14E15 ,Computer Science::Computer Science and Game Theory ,Mathematics::Commutative Algebra ,FOS: Mathematics ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra - Abstract
This paper gives a survey on a valuation theoretical approach to local uniformization in positive characteristic, the model theory of valued fields in positive characteristic, and their connection with the valuation theoretical phenomenon of defect., 79 pages; in: Resolution of Singularities - A Research Textbook in Tribute to Oscar Zariski. Herwig Hauser, Joseph Lipman, Frans Oort, Adolfo Quiros (eds.)
- Published
- 2010
50. Elimination of Ramification I: The Generalized Stability Theorem
- Author
-
Franz-Viktor Kuhlmann
- Subjects
Model theory ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Ramification (botany) ,010102 general mathematics ,Field (mathematics) ,Extension (predicate logic) ,Function (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,0103 physical sciences ,Calculus ,FOS: Mathematics ,Primitive element theorem ,010307 mathematical physics ,Transcendental number ,0101 mathematics ,Uniformization (set theory) ,12J10 (Primary), 13A18, 12L12 (Secondary), 14B05 ,Mathematics - Abstract
We prove a general version of the "Stability Theorem": if $K$ is a valued field such that the ramification theoretical defect is trivial for all of its finite extensions, and if $F|K$ is a finitely generated (transcendental) extension of valued fields for which equality holds in the Abhyankar inequality, then the defect is also trivial for all finite extensions of $F$. This theorem is applied to eliminate ramification in such valued function fields. It has applications to local uniformization and to the model theory of valued fields in positive characteristic., Comment: 31 pages
- Published
- 2010
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