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TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS.

The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the compu...

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Published in:Journal of Symbolic Logic Vol. 82; no. 1; pp. 1 - 26
Main Authors: Marker, David, Miller, Russell
Format: Article
Published: Cambridge University Press Mar2017
Subjects:
Spectrum analysis
Turing (Computer program language)
Differential fields
Computable functions
Differential operators
Online Access:View this record in EBSCOhost
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        atl: TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS.
      aug:
        au:
          Marker, David
          Miller, Russell
        affil:
          DEPARTMENT OF MATHEMATICS STATISTICS, & COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO CHICAGO, IL, USA E-mail:
          DEPARTMENT OF MATHEMATICS QUEENS COLLEGE - CUNY 65-30 KISSENA BLVD. QUEENS, NY 11367, USA and PH.D. PROGRAMS IN MATHEMATICS AND COMPUTER SCIENCE CUNY GRADUATE CENTER 365 FIFTH AVENUE NEW YORK, NY 10016, USA E-mail: URL: http://qcpages.qc.cuny.edu/∼rmiller/
      su:
        Spectrum analysis
        Turing (Computer program language)
        Differential fields
        Computable functions
        Differential operators
      sug:
        subj:
          Spectrum analysis
          Turing (Computer program language)
          Differential fields
          Computable functions
          Differential operators
      keyword:
        03C10
        12H05
        computability
        computable model theory
        differentially closed fields
        model theory
        Primary 03D45
        Secondary 03C57
        Turing degree spectrum
      ab: The degree spectrum of a countable structure is the set of all Turing degrees of presentations of that structure. We show that every nonlow Turing degree lies in the spectrum of some differentially closed field (of characteristic 0, with a single derivation) whose spectrum does not contain the computable degree 0. Indeed, this is an equivalence, for we also show that if this spectrum contained a low degree, then it would contain the degree 0. From these results we conclude that the spectra of differentially closed fields of characteristic 0 are exactly the jump-preimages of spectra of automorphically nontrivial graphs.
      pubtype: Academic Journal
      doctype: Article
      src: R
    language: English
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