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A valuation theoretic characterization of recursively saturated real closed fields [article]

Paola D'Aquino, Salma Kuhlmann, Karen Lange
2012 arXiv   pre-print
We give a valuation theoretic characterization for a real closed field to be recursively saturated.  ...  Our result extends the characterization of Harnik and Ressayre hr for a divisible ordered abelian group to be recursively saturated.  ...  Research visits of the first and second authors  ... 
arXiv:1212.6842v1 fatcat:6qikjwjbinaulhhcnh3cxmcbia

A VALUATION THEORETIC CHARACTERIZATION OF RECURSIVELY SATURATED REAL CLOSED FIELDS

PAOLA D'AQUINO, SALMA KUHLMANN, KAREN LANGE
2015 Journal of Symbolic Logic (JSL)  
We give a valuation theoretic characterization for a real closed field to be recursively saturated.  ...  Our result extends the characterization of Harnik and Ressayre [7] for a divisible ordered abelian group to be recursively saturated.  ...  Recursively saturated real closed fields. We now give an analogous characterization of recursively saturated real closed fields. We need the following definition. Definition 5.1.  ... 
doi:10.1017/jsl.2014.21 fatcat:gl7ktdpoavbnncrt24au332lru

On the value group of a model of Peano Arithmetic

Merlin Carl, Paola D'Aquino, Salma Kuhlmann
2017 Forum mathematicum  
On the other hand, an algebraic characterization of recursively saturated real closed fields (of any cardinality) is given in [DKL] .  ...  In particular, it is shown that the value group of a recursively saturated real closed field is recursively saturated [DKL, Theorem 5.2 ].  ... 
doi:10.1515/forum-2015-0226 fatcat:xyy2ngudk5cwjaoxhyhzn3ffoi

Representing Scott sets in algebraic settings [article]

Alf Dolich, Julia Knight, Karen Lange, David Marker
2014 arXiv   pre-print
We prove that for every Scott set S there are S-saturated real closed fields and models of Presburger arithmetic.  ...  D'Aquino, Kuhlmann and Lange [3] gave a valuation-theoretic characterization of recursively saturated real closed fields.  ...  Every real closed field K has a natural valuation for which the valuation ring is O = {x : |x| < n for some n ∈ N}.  ... 
arXiv:1407.5612v1 fatcat:3sbukd25ljal3p7wqav6lqf4j4

Representing Scott sets in algebraic settings

Alf Dolich, Julia F. Knight, Karen Lange, David Marker
2015 Archive for Mathematical Logic  
D'Aquino, Kuhlmann and Lange [3] gave a valuation-theoretic characterization of recursively saturated real closed fields.  ...  Every real closed field K has a natural valuation for which the valuation ring is O = {x : |x| < n for some n ∈ N}.  ... 
doi:10.1007/s00153-015-0431-1 fatcat:p4qzx5jvpfgdhpyykerwsrwmom

Page 3915 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
These canonical forms are a finite union of Swiss cheeses for algebraically closed valued fields, and a finite union of concentric annuli for real closed valued fields.  ...  (Valuations are, by assumption, non- trivial, and, in the real closed case, convex.)  ... 

Characterizing reduced Witt rings. II

Thomas Craven
1979 Pacific Journal of Mathematics  
We have recently given a recursive construction of all reduced Witt rings of fields with finitely many places into the real numbers.  ...  In this paper we extend the construction to include all reduced Witt rings of fields. We then demonstrate how this recursion process can be used to prove facts about these rings.  ...  The Pacific Journal of Mathematics is issued monthly as of January 1966. Regular subscription rate: $72.00 a year (6 Vols., 12 issues).  ... 
doi:10.2140/pjm.1979.80.341 fatcat:aziuhtvdebcjtbrgvlmm3bvdfq

Pairs of theories satisfying a Mordell–Lang condition

Alexi Block Gorman, Philipp Hieronymi, Elliot Kaplan, Alexi Block Gorman
2020 Fundamenta Mathematicae  
ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars.  ...  The second is whether there is a subfield K of a real closed field that is not real closed, yet every open set definable in the expansion of the real field by K is semialgebraic.  ...  The first author was partially supported by a DOE GAANN fellowship. The second author was partially supported by NSF grant DMS-1654725.  ... 
doi:10.4064/fm857-1-2020 fatcat:rkdv4pe4xrdr5hc6v73q4g4ley

Open induction in a bounded arithmetic for TC^0 [article]

Emil Jeřábek
2014 arXiv   pre-print
only TC^0-computable objects, i.e., in a theory of bounded arithmetic corresponding to TC^0.  ...  The elementary arithmetic operations +,·,< on integers are well-known to be computable in the weak complexity class TC^0, and it is a basic question what properties of these operations can be proved using  ...  The ω-saturation of D implies that every Dedekind cut on Q is realized by an element of F , hence in fact k = R, which is a real-closed field.  ... 
arXiv:1404.7435v1 fatcat:65o2triwynfjfeannpmsjegjvi

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

Alessandro Berarducci, Philip Ehrlich, Salma Kuhlmann
2017 Oberwolfach Reports  
Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H.  ...  Conway's field No of surreal numbers, have been lately discovered and exploited.  ...  Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268,  ... 
doi:10.4171/owr/2016/60 fatcat:afef7i27qrhsrhfm67hz6h4lqm

Dimension in the realm of transseries [article]

Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven
2017 arXiv   pre-print
Let T be the differential field of transseries. We establish some basic properties of the dimension of a definable subset of T^n, also in relation to its codimension in the ambient space T^n.  ...  The case of dimension 0 is of special interest, and can be characterized both in topological terms (discreteness) and in terms of the Herwig-Hrushovski-Macpherson notion of co-analyzability.  ...  Since "Liouville closed" includes "real closed", the ordering (and thus the valuation ring) of any model of T nl is definable in the underlying differential field of the model.  ... 
arXiv:1607.07173v2 fatcat:lquzgkaymrbvdf2zul2azlyzpy

Pairs of Theories Satisfying a Mordell-Lang Condition [article]

Alexi Block Gorman, Philipp Hieronymi, Elliot Kaplan
2018 arXiv   pre-print
ones, such as pairs consisting of a real closed field and a pseudo real closed subfield, and pairs of vector spaces with different fields of scalars.  ...  This paper proposes a new setup for studying pairs of structures.  ...  Real closed field with a predicate for a pseudo real closed subfield In this section, we consider a real closed field with a predicate for a dense pseudo real closed subfield with n orderings where n ≥  ... 
arXiv:1806.00030v1 fatcat:cgdmwoxfqzcunbsyz67fv6scjq

Denseness results in the theory of algebraic fields [article]

Sylvy Anscombe, Philip Dittmann, Arno Fehm
2019 arXiv   pre-print
We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property.  ...  The authors would like to thank the IHP for funding and hospitality, and the organizers of the trimester 'Model theory, combinatorics and valued fields' for the invitation.  ...  Acknowledgements Part of this work was done while all three authors were guests of the Institut Henri Poincaré.  ... 
arXiv:1909.12188v1 fatcat:hmrz4prb2fdd7hf2dpkv2cgnli

Decidability via the tilting correspondence [article]

Kartas Konstantinos
2021 arXiv   pre-print
hull of 𝔽_p((t)).  ...  This applies to show that the fields ℚ_p(p^1/p^∞) and ℚ_p(ζ_p^∞) are (existentially) decidable relative to the perfect hull of 𝔽_p((t)) and ℚ_p^ab is (existentially) decidable relative to the perfect  ...  A great mathematical debt is owed to E. Hrushovski from whom I learned many of the crucial ideas in this work. I would like to thank J.  ... 
arXiv:2001.04424v4 fatcat:p2jnnkmgzjgvjnqtidna4bb4vi

SV-Rings and SV-Porings

Niels Schwartz
2010 Annales de la Faculté des Sciences de Toulouse  
Cherlin and Dickmann called such rings real closed; here they will be called real closed valuation rings. [35] is a study of real closed valuation rings vs. the larger class of real closed domains.  ...  Most factor domains of rings of continuous functions are not valuation rings; if they are valuation rings then they are convex subrings of real closed fields.  ...  A. Then there is an element c ∈ A such that a s k ≡ c (mod A s · p), say s l · (a − c · s k ) ∈ p. Since s ∈ p it follows that a − c · s k ∈ p, and therefore 0 < |a − c · s k | m < s for every m 1.  ... 
doi:10.5802/afst.1280 fatcat:o3jfwpppvbe6xkqpcb46t6tmwi
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