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Journal articles on the topic 'Valued fields'

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Published: 4 June 2021
Last updated: 28 September 2024

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1

Harrison-Trainor, Matthew. "Computable valued fields." Archive for Mathematical Logic 57, no. 5-6 (September 15, 2017): 473–95. http://dx.doi.org/10.1007/s00153-017-0589-9.

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2

Ershov, Yu L. "*-Extremal valued fields." Siberian Mathematical Journal 50, no. 6 (November 2009): 1007–10. http://dx.doi.org/10.1007/s11202-009-0111-7.

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3

Ershov, Yu L. "Extremal Valued Fields." Algebra and Logic 43, no. 5 (September 2004): 327–30. http://dx.doi.org/10.1023/b:allo.0000044281.72007.d0.

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4

Ershov, Yu L. "Stable valued fields." Algebra and Logic 46, no. 6 (November 2007): 385–98. http://dx.doi.org/10.1007/s10469-007-0038-7.

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5

Khanduja, Sudesh K. "On value groups and residue fields of some valued function fields." Proceedings of the Edinburgh Mathematical Society 37, no. 3 (October 1994): 445–54. http://dx.doi.org/10.1017/s0013091500018897.

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Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).
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6

Prestel, Alexander, and Cydara C. Ripoll. "Integral-valued rational functions on valued fields." Manuscripta Mathematica 73, no. 1 (December 1991): 437–52. http://dx.doi.org/10.1007/bf02567653.

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7

Guzy, Nicolas. "0-D-valued fields." Journal of Symbolic Logic 71, no. 2 (June 2006): 639–60. http://dx.doi.org/10.2178/jsl/1146620164.

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AbstractIn [12]. T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.
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8

Pal, Koushik. "Multiplicative valued difference fields." Journal of Symbolic Logic 77, no. 2 (June 2012): 545–79. http://dx.doi.org/10.2178/jsl/1333566637.

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AbstractThe theory of valued difference fields (K, σ, υ,) depends on how the valuation υ interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where υ(σ(x)) = υ(x) for all x Є K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where υ(σ(x)) > nυ(x) for all x Є K× with υ(x) > 0 and n Є ℕ, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where υ(σ(x)) = ρ · υ(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for this theory.
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9

JAHNKE, FRANZISKA, PIERRE SIMON, and ERIK WALSBERG. "DP-MINIMAL VALUED FIELDS." Journal of Symbolic Logic 82, no. 1 (March 2017): 151–65. http://dx.doi.org/10.1017/jsl.2016.15.

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10

Jahnke, Franziska, and Pierre Simon. "NIP henselian valued fields." Archive for Mathematical Logic 59, no. 1-2 (June 29, 2019): 167–78. http://dx.doi.org/10.1007/s00153-019-00685-8.

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11

Guzy, Nicolas. "Valued Fields withKCommuting Derivations." Communications in Algebra 34, no. 12 (December 2006): 4269–89. http://dx.doi.org/10.1080/00927870600877985.

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12

Chipchakov, I. D. "Henselian Valued Stable Fields." Journal of Algebra 206, no. 1 (August 1998): 344–69. http://dx.doi.org/10.1006/jabr.1997.7396.

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13

Pynn-Coates, Nigel. "Newtonian valued differential fields with arbitrary value group." Communications in Algebra 47, no. 7 (February 6, 2019): 2766–76. http://dx.doi.org/10.1080/00927872.2018.1539173.

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14

Khanduja, Sudesh K., and Usha Garg. "Residue fields of valued function fields of conics." Proceedings of the Edinburgh Mathematical Society 36, no. 3 (October 1993): 469–78. http://dx.doi.org/10.1017/s0013091500018551.

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Suppose that K is a function field of a conic over a subfield K0. Let v0 be a valuation of K0 with residue field k0 of characteristic ≠2. Let v be an extension of v0 to K having residue field k. It has been proved that either k is an algebraic extension of k0 or k is a regular function field of a conic over a finite extension of k0. This result can also be deduced from the genus inequality of Matignon (cf. [On valued function fields I, Manuscripta Math. 65 (1989), 357–376]) which has been proved using results about vector space defect and methods of rigid analytic geometry. The proof given here is more or less self-contained requiring only elementary valuation theory.
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15

Azgin, Salih. "Valued fields with contractive automorphism and Kaplansky fields." Journal of Algebra 324, no. 10 (November 2010): 2757–85. http://dx.doi.org/10.1016/j.jalgebra.2010.08.003.

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16

Warner, Seth. "Finite Extensions of Valued Fields." Canadian Mathematical Bulletin 29, no. 1 (March 1, 1986): 64–69. http://dx.doi.org/10.4153/cmb-1986-012-x.

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AbstractA corollary of the main result is that if L is a finite-dimensional Galois extension of a field K and if w is a valuation of L extending a valuation v of K, then K is closed in L if and only if all valuations of L extending v are dependent. A further consequence is a generalization of Ostrowski's criterion for a real-valued valuation to be henselian.
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17

Khassa, Ramneek, and Sudesh K. Khanduja. "Subfields of henselian valued fields." Colloquium Mathematicum 120, no. 1 (2010): 157–63. http://dx.doi.org/10.4064/cm120-1-12.

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18

CHIPCHAKOV, IVAN. "Henselian discrete valued stable fields." Turkish Journal of Mathematics 46, no. 5 (January 1, 2022): 1735–48. http://dx.doi.org/10.55730/1300-0098.3229.

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19

Nart, Enric. "Key polynomials over valued fields." Publicacions Matemàtiques 64 (January 1, 2020): 195–232. http://dx.doi.org/10.5565/publmat6412009.

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20

Azgin, Salih, Franz-Viktor Kuhlmann, and Florian Pop. "Characterization of extremal valued fields." Proceedings of the American Mathematical Society 140, no. 5 (May 1, 2012): 1535–47. http://dx.doi.org/10.1090/s0002-9939-2011-11020-7.

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21

Ben Yaacov, Itaï. "Tensor products of valued fields." Bulletin of the London Mathematical Society 47, no. 1 (December 4, 2014): 42–46. http://dx.doi.org/10.1112/blms/bdu092.

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22

Mathiak, Karl. "Completion of valued skew fields." Journal of Algebra 150, no. 2 (August 1992): 257–70. http://dx.doi.org/10.1016/s0021-8693(05)80030-4.

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23

Prestel, Alexander. "Positive elimination in valued fields." manuscripta mathematica 123, no. 1 (April 11, 2007): 95–103. http://dx.doi.org/10.1007/s00229-007-0087-1.

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24

Chernikov, Artem, and Martin Hils. "Valued difference fields and NTP2." Israel Journal of Mathematics 204, no. 1 (June 25, 2014): 299–327. http://dx.doi.org/10.1007/s11856-014-1094-z.

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25

Popescu, Dorin. "Algebraic extensions of valued fields." Journal of Algebra 108, no. 2 (July 1987): 513–33. http://dx.doi.org/10.1016/0021-8693(87)90114-1.

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26

Bhaskaran, R., and V. Karunakaran. "Analytic functions over valued fields." International Journal of Mathematics and Mathematical Sciences 13, no. 2 (1990): 247–52. http://dx.doi.org/10.1155/s0161171290000370.

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LetKbe a non-archimedean, non-trivially (rank 1) valued complete field.B,B0denote the closed and open unit ball ofKrespectively. Necessary and sufficient conditions for analytic functions defined onB,B0with values inKto be injective, necessary and sufficient conditions for fixed points, the problem of subordination are studied in this paper.
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27

Pasol, Vicentiu, Angel Popescu, and Nicolae Popescu. "Spectral norms on valued fields." Mathematische Zeitschrift 238, no. 1 (September 2001): 101–14. http://dx.doi.org/10.1007/pl00004895.

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28

Green, B., M. Matignon, and F. Pop. "On valued function fields I." Manuscripta Mathematica 65, no. 3 (September 1989): 357–76. http://dx.doi.org/10.1007/bf01303043.

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29

Ribenboim, P. "Some Examples of Valued Fields." Journal of Algebra 173, no. 3 (May 1995): 668–78. http://dx.doi.org/10.1006/jabr.1995.1108.

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30

Sturm, Thomas. "Linear Problems in Valued Fields." Journal of Symbolic Computation 30, no. 2 (August 2000): 207–19. http://dx.doi.org/10.1006/jsco.1999.0303.

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31

HASKELL, DEIRDRE, and YOAV YAFFE. "GANZSTELLENSÄTZE IN THEORIES OF VALUED FIELDS." Journal of Mathematical Logic 08, no. 01 (June 2008): 1–22. http://dx.doi.org/10.1142/s0219061308000695.

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The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make it dependent on the relevant class of valued fields. We apply our framework to algebraically closed valued fields, model complete theories of difference and differential valued fields, and real closed valued fields.
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32

Chipchakov, I. D. "On the residue fields of Henselian valued stable fields." Journal of Algebra 319, no. 1 (January 2008): 16–49. http://dx.doi.org/10.1016/j.jalgebra.2007.08.034.

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33

ANSCOMBE, SYLVY, and FRANZ-VIKTOR KUHLMANN. "NOTES ON EXTREMAL AND TAME VALUED FIELDS." Journal of Symbolic Logic 81, no. 2 (June 2016): 400–416. http://dx.doi.org/10.1017/jsl.2016.6.

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AbstractWe 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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34

Touchard, Pierre. "Transfer Principles in Henselian Valued Fields." Bulletin of Symbolic Logic 27, no. 2 (June 2021): 222–23. http://dx.doi.org/10.1017/bsl.2021.31.

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AbstractIn this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye.These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups.Abstract prepared by Pierre Touchard.E-mail: pierre.pa.touchard@gmail.comURL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9
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35

Chipchakov, I. D. "Henselian valued quasi-local fields with totally indivisible value groups." Communications in Algebra 27, no. 7 (January 1999): 3093–108. http://dx.doi.org/10.1080/00927879908826612.

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36

Mahboub, W., A. Mansour, and M. Spivakovsky. "On common extensions of valued fields." Journal of Algebra 584 (October 2021): 1–18. http://dx.doi.org/10.1016/j.jalgebra.2021.04.027.

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37

El Fadil, L., M. Boulagouaz, and A. Deajim. "A Dedekind Criterion over Valued Fields." Siberian Mathematical Journal 62, no. 5 (September 2021): 868–75. http://dx.doi.org/10.1134/s0037446621050098.

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38

El Fadil, L., M. Boulagouaz, and A. Deajim. "A Dedekind criterion over valued fields." Sibirskii matematicheskii zhurnal 62, no. 5 (August 30, 2021): 1073–83. http://dx.doi.org/10.33048/smzh.2021.62.509.

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39

Hils, Martin, Moshe Kamensky, and Silvain Rideau. "Imaginaries in separably closed valued fields." Proceedings of the London Mathematical Society 116, no. 6 (February 1, 2018): 1457–88. http://dx.doi.org/10.1112/plms.12116.

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40

Popescu, Angel, Nicolae Popescu, and Alexandru Zaharescu. "Metric invariants over Henselian valued fields." Journal of Algebra 266, no. 1 (August 2003): 14–26. http://dx.doi.org/10.1016/s0021-8693(03)00379-x.

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41

Lee, Junguk. "Hyperfields, truncated DVRs, and valued fields." Journal of Number Theory 212 (July 2020): 40–71. http://dx.doi.org/10.1016/j.jnt.2019.10.019.

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42

Cluckers, Raf, and Deirdre Haskell. "Grothendieck Rings of ℤ-Valued Fields." Bulletin of Symbolic Logic 7, no. 2 (June 2001): 262–69. http://dx.doi.org/10.1017/s1079898600005849.

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AbstractWe prove the triviality of the Grothendieck ring of a ℤ-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K2 to itself minus a point. When we specialize to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.
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43

van der Linden, F. J. "Integer valued polynomials over function fields." Indagationes Mathematicae (Proceedings) 91, no. 3 (September 1988): 293–308. http://dx.doi.org/10.1016/s1385-7258(88)80009-x.

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44

FLENNER, JOSEPH, and VINCENT GUINGONA. "CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS." Journal of Symbolic Logic 79, no. 01 (March 2014): 154–70. http://dx.doi.org/10.1017/jsl.2013.16.

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Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.
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45

KOWALSKI, PIOTR, and SERGE RANDRIAMBOLOLONA. "STRONGLY MINIMAL REDUCTS OF VALUED FIELDS." Journal of Symbolic Logic 81, no. 2 (June 2016): 510–23. http://dx.doi.org/10.1017/jsl.2015.61.

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AbstractWe prove that if a strongly minimal nonlocally modular reduct of an algebraically closed valued field of characteristic 0 contains +, then this reduct is bi-interpretable with the underlying field.
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46

HONG, JIZHAN. "SEPARABLY CLOSED VALUED FIELDS: QUANTIFIER ELIMINATION." Journal of Symbolic Logic 81, no. 3 (August 12, 2016): 887–900. http://dx.doi.org/10.1017/jsl.2015.62.

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AbstractIt 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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47

CHERNIKOV, ARTEM, and PIERRE SIMON. "HENSELIAN VALUED FIELDS AND inp-MINIMALITY." Journal of Symbolic Logic 84, no. 4 (August 29, 2019): 1510–26. http://dx.doi.org/10.1017/jsl.2019.56.

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AbstractWe prove that every ultraproduct of p-adics is inp-minimal (i.e., of burden 1). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic 0 in the RV language.
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48

Haran, Dan, and Helmut Völklein. "Galois groups over complete valued fields." Israel Journal of Mathematics 93, no. 1 (December 1996): 9–27. http://dx.doi.org/10.1007/bf02761092.

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49

Nanda, Sudarsan. "Fuzzy linear spaces over valued fields." Fuzzy Sets and Systems 42, no. 3 (August 1991): 351–54. http://dx.doi.org/10.1016/0165-0114(91)90113-5.

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50

Hong, Jizhan. "Separably closed valued fields: Immediate expansions." Israel Journal of Mathematics 216, no. 2 (October 2016): 811–31. http://dx.doi.org/10.1007/s11856-016-1428-0.

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