15 results on '"Franz-Viktor Kuhlmann"'
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2. Valuations on rational function fields that are invariant under permutation of the variables
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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.
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- 2016
3. Eliminating Tame Ramification: generalizations of Abhyankar's Lemma
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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.
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- 2019
- Full Text
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4. SEPARABLY CLOSED VALUED FIELDS: QUANTIFIER ELIMINATION
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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 ${\cal L} = \{ + , - , \times ,0,1\} \cup \{ {b_1}, \ldots ,{b_e}\} \cup {\{ {\lambda _{e,j}}(x;{b_1}, \ldots ,{b_e})\} _{0 \le j < {p^e}}} \cup \{ |\}$.
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- 2016
5. On maximal immediate extensions of valued fields
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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
6. NOTES ON EXTREMAL AND TAME VALUED FIELDS
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Sylvy Anscombe and Franz-Viktor Kuhlmann
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G110 ,Logic ,010102 general mathematics ,Mathematics - Logic ,Commutative Algebra (math.AC) ,Mathematics - Commutative Algebra ,01 natural sciences ,Philosophy ,Primary 12J20, Secondary 12J10 ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Logic (math.LO) - Abstract
We extend the characterization of extremal valued fields given in [2] 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 finitep-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 [8] 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 ℵ1saturated valued field the valuation is a composition of extremal valuations of rank 1.
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- 2016
7. The model theory of separably tame valued fields
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Koushik Pal and Franz-Viktor Kuhlmann
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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
8. Counting the number of distinct distances of elements in valued field extensions
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Franz-Viktor Kuhlmann and Anna Blaszczok
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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.
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- 2017
9. The algebra and model theory of tame valued fields
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Franz-Viktor Kuhlmann
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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.
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- 2014
10. The relative approximation degree in valued function fields
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Franz-Viktor Kuhlmann and Izabela Vlahu
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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.
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- 2013
11. Elimination of Ramification I: The Generalized Stability Theorem
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Franz-Viktor Kuhlmann
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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
- Full Text
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12. Places of algebraic function fields in arbitrary characteristic
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Franz-Viktor Kuhlmann
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Algebraic function field ,Discrete mathematics ,Zariski tangent space ,Zariski topology ,Mathematics(all) ,Spectrum of a ring ,Places of algebraic function fields ,General Mathematics ,010102 general mathematics ,12J10 ,Algebraic variety ,Field (mathematics) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Generic point ,Large fields ,Zariski space ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,0101 mathematics ,Irreducible component ,Mathematics - Abstract
We consider the Zariski space of all places of an algebraic function field $F|K$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zero-dimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$-rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasi-compact and that it is a spectral space., Comment: 27 pages
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- 2010
- Full Text
- View/download PDF
13. ADDITIVE POLYNOMIALS AND THEIR ROLE IN THE MODEL THEORY OF VALUED FIELDS
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Franz-Viktor Kuhlmann
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Discrete mathematics ,Ring (mathematics) ,Discrete orthogonal polynomials ,010102 general mathematics ,Free module ,Field (mathematics) ,01 natural sciences ,Classical orthogonal polynomials ,Difference polynomials ,0103 physical sciences ,Algebraic function ,010307 mathematical physics ,0101 mathematics ,Mathematics ,Variable (mathematics) - Abstract
We discuss the role of additive polynomials and p-polynomials in the theory of valued fields of positive characteristic and in their model theory. We outline the basic properties of rings of additive polynomials and discuss properties of valued fields of positive characteristic as modules over such rings. We prove the existence of Frobenius-closed bases of algebraic function fields F|K in one variable and deduce that F/K is a free module over the ring of additive polynomials with coefficients inK. Finally, we prove that every minimal purely wild extension of a henselian valued field is generated by a p-polynomial.
- Published
- 2006
14. Valuation theory of exponential Hardy fields I
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Salma Kuhlmann and Franz-Viktor Kuhlmann
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Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Valuation theory ,01 natural sciences ,Exponential function ,Exponential growth ,Bounded function ,0103 physical sciences ,010307 mathematical physics ,Hardy field ,0101 mathematics ,Valuation (measure theory) ,ddc:510 ,Power function ,Mathematics ,Analytic function - Abstract
We describe the valuation theoretic properties of the Hardy fields associated to models of $T(\exp)$ , where T is the theory of a polynomially bounded o-minimal expansion of the reals and $\exp$ is the real exponential function. We deduce that $T(\exp)$ has levels with parameters and is exponentially bounded. We establish a maximality property of $H(\mathbb{R}_{\rm an, powers})$ , the Hardy field of the expansion by the restricted analytic functions and power functions.
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- 2003
15. On generalized series fields and exponential-logarithmic series fields with derivations
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Mickaël Matusinski, Antonio Campillo (Universidad de Valladolid, Spain), Franz-Viktor Kuhlmann (University of Saskatchewan, Saskatoon, Canada), Bernard Teissier (Institut de Mathématiques de Jussieu, Paris, France), Équipe Géométrie, Institut de Mathématiques de Bordeaux (IMB), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Université de Bordeaux (UB)-Institut Polytechnique de Bordeaux (Bordeaux INP)-Centre National de la Recherche Scientifique (CNRS)
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[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC] ,010102 general mathematics ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,01 natural sciences - Abstract
International audience; We survey some important properties of fields of generalized series and of exponential-logarithmic series, with particular emphasis on their possible differential structure, based on a joint work of the author with S. Kuhlmann [KM12b,KM11].
- Published
- 2014
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