Quantifier elimination for quasi-real closed fields

Mickaël Matusinski; Simon Müller

doi:10.5802/crmath.169

Logique mathématique

Quantifier elimination for quasi-real closed fields

Mickaël Matusinski ¹ ; Simon Müller ²

¹ Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
² Universität Konstanz, 78467 Konstanz, Germany

Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 291-295.

Résumé
Abstract

Nous prouvons l’élimination des quantificateurs pour la théorie des corps quasi-réels clos munis d’une valuation compatible. Cela reprend et unifie les mêmes résultats connus pour les corps algébriquement clos et les corps réels clos.

We prove quantifier elimination for the theory of quasi-real closed fields with a compatible valuation. This unifies the same known results for algebraically closed valued fields and real closed valued fields.

Reçu le : 2020-05-26
Accepté le : 2020-12-13
Publié le : 2021-04-20

DOI : 10.5802/crmath.169

Classification : 03C10, 03C64, 12J10, 12J15, 12L12

Affiliations des auteurs :

Mickaël Matusinski ¹ ; Simon Müller ²

¹ Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France
² Universität Konstanz, 78467 Konstanz, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Micka\"el Matusinski and Simon M\"uller},
     title = {Quantifier elimination for quasi-real closed fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {291--295},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {3},
     year = {2021},
     doi = {10.5802/crmath.169},
     language = {en},
}

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Mickaël Matusinski; Simon Müller. Quantifier elimination for quasi-real closed fields. Comptes Rendus. Mathématique, Volume 359 (2021) no. 3, pp. 291-295. doi : 10.5802/crmath.169. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.169/

Bibliographie
Cité par

[1] Emil Artin; Otto Schreier Algebraische Konstruktion reeller Körper, Abh. Math. Semin. Univ. Hamb., Volume 5 (1927) no. 1, pp. 85-99 | DOI | Zbl

[2] Reinhold Baer Über nicht-archimedisch geordnete Körper. (Beiträge zur Algebra 5.)., Volume 8, 1927, pp. 3-13 | Zbl

[3] Gregory Cherlin; Max A. Dickmann Real closed rings II, Ann. Pure Appl. Logic, Volume 25 (1983) no. 3, pp. 213-231 | DOI | MR | Zbl

[4] Charles N. Delzell; Alexander Prestel Mathematical logic and model theory. A brief introduction, Universitext, Springer, 2011, x+193 pages | Zbl

[5] Ido Efrat Valuations, orderings, and Milnor $K$ -theory, Mathematical Surveys and Monographs, 124, American Mathematical Society, 2006, xiv+288 pages | DOI | MR | Zbl

[6] Antonio J. Engler; Alexander Prestel Valued fields, Springer Monographs in Mathematics, Springer, 2005, x+205 pages | Zbl

[7] Syed M. Fakhruddin Quasi-ordered fields, J. Pure Appl. Algebra, Volume 45 (1987) no. 3, pp. 207-210 | DOI | MR | Zbl

[8] Wolfgang Krull Allgemeine Bewertungstheorie, J. Reine Angew. Math., Volume 167 (1932), pp. 160-196 | DOI | MR | Zbl

[9] Salma Kuhlmann; Mickaël Matusinski; Françoise Point The Valuation Difference Rank of a Quasi-Ordered Difference Field, Groups, Modules, and Model Theory – Surveys and Recent Developments : In Memory of Rüdiger Göbel (Manfred Droste; László Fuchs; Brendan Goldsmith; Lutz Strüngmann, eds.), Springer, 2017, pp. 399-414 | DOI | Zbl

[10] Salma Kuhlmann; Simon Müller Compatibility of Quasi-Orderings and Valuations: A Baer–Krull Theorem for Quasi-Ordered Rings, Order, Volume 36 (2019) no. 2, pp. 249-269 | DOI | MR | Zbl

[11] Abraham Robinson Complete theories, Studies in Logic and the Foundations of Mathematics, North-Holland, 1956 | Zbl

[12] Joseph Shipman Improving the fundamental theorem of algebra, Math. Intell., Volume 29 (2007) no. 4, pp. 9-14 | DOI | MR | Zbl

[13] Oscar Zariski; Pierre Samuel Commutative algebra. Vol. II, Graduate Texts in Mathematics, 29, Springer, 1975, x+414 pages (Reprint of the 1960 edition) | MR | Zbl

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