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TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS

Published online by Cambridge University Press:  21 March 2017

DAVID MARKER  and
RUSSELL MILLER
DAVID MARKER
Affiliation:
DEPARTMENT OF MATHEMATICS STATISTICS, & COMPUTER SCIENCE UNIVERSITY OF ILLINOIS AT CHICAGO CHICAGO, IL, USAE-mail: marker@math.uic.edu
RUSSELL MILLER
Affiliation:
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, USAE-mail: russell.miller@qc.cuny.eduURL: http://qcpages.qc.cuny.edu/∼rmiller/
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Abstract

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.

Keywords

Primary 03D45Secondary 03C5712H0503C10computabilitycomputable model theorydifferentially closed fieldsmodel theoryTuring degree spectrum
Type
Articles
Information
The Journal of Symbolic Logic , Volume 82 , Issue 1 , March 2017 , pp. 1 - 25
Copyright
Copyright © The Association for Symbolic Logic 2017 

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References

REFERENCES

Ash, C. J., Jockusch, C. G. Jr., and Knight, J. F., Jumps of orderings . Transactions of the American Mathematical Society, vol. 319 (1990), no. 2, pp. 573–599.Google Scholar
Blum, L., Generalized algebraic theories: A model theoretic approach , Ph.D. thesis, Massachusetts Institute of Technology, 1968.Google Scholar
Blum, L., Differentially closed fields: A model-theoretic tour , Contributions to Algebra (collection of papers dedicated to Ellis Kolchin) (Bass, H., Cassidy, P., and Kovacic, J., editors), Academic Press, New York, 1977, pp. 37–61.Google Scholar
Downey, R. and Jockusch, C. Jr., Every low Boolean algebra is isomorphic to a recursive one . Proceedings of the American Mathematical Society, vol. 122 (1994), pp. 871–880.Google Scholar
Downey, R. and Knight, J. F., Orderings with αth jump degree 0(α) . Proceedings of the American Mathematical Society, vol. 114 (1992), no. 2, pp. 545–552.Google Scholar
Frolov, A., Harizanov, V., Kalimullin, I., Kudinov, O., and Miller, R., Degree spectra of high n and non-low n degrees . Journal of Logic and Computation, vol. 22 (2012), no. 4, pp. 755–777.CrossRefGoogle Scholar
Harrington, L., Recursively presentable prime models, this Journal, vol. 39 (1974), no. 2, pp. 305–309.Google Scholar
Hirschfeldt, D. R., Khoussainov, B., Shore, R. A., and Slinko, A. M., Degree spectra and computable dimensions in algebraic structures . Annals of Pure and Applied Logic, vol. 115 (2002), pp. 71–113.Google Scholar
Hrushovski, E. and Itai, M., On model complete differential fields . Transactions of the American Mathematical Society, vol. 355 (2003), no. 11, pp. 4267–4296.Google Scholar
Hrushovski, E. and Sokolović, Z., Minimal subsets of differentially closed fields, preprint from the early 1990s.Google Scholar
Jockusch, C. G. and Soare, R. I., Degrees of orderings not isomorphic to recursive linear orderings . Annals of Pure and Applied Logic, vol. 52 (1991), pp. 39–64.Google Scholar
Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 1034–1042.Google Scholar
Knight, J. F. and Stob, M., Computable Boolean algebras, this Journal, vol. 65 (2000), no. 4, pp. 1605–1623.Google Scholar
Marker, D. and Kernels, M., Connections Between Model Theory and Algebraic and Analytic Geometry , Quaderni di Matematica, vol. 6, Dipartimento di Matematica II Università di Napoli, Caserta, 2000, pp. 1–21.Google Scholar
Marker, D., Model theory of differential fields , Model Theory of Fields (Marker, D., Messmer, M., and Pillay, A., editors), ASL Lecture Notes in Logic, vol. 5, A.K. Peters, Ltd., Wellesley, MA, 2006, pp. 41–109.Google Scholar
Miller, R. G., Computable fields and Galois theory , Notices of the AMS, vol. 55 (2008), no. 7, pp. 798–807.Google Scholar
Miller, R., Ovchinnikov, A., and Trushin, D., Computing constraint sets for differential fields . Journal of Algebra, vol. 407 (2014), pp. 316–357.Google Scholar
Miller, R., Poonen, B., Schoutens, H., and Shlapentokh, A., A computable functor from graphs to fields, submitted for publication.Google Scholar
Montalbán, A., Mathematical Theory and Computational Practice: Fifth Conference on Computability in Europe, CiE 2009 (Ambos-Spies, K., Löwe, B., and Merkle, W., editors), Lecture Notes in Computer Science, vol. 5635, Springer-Verlag, Berlin, 2009.Google Scholar
Nagloo, J. and Pillay, A., On algebraic relations between solutions of a generic Painlevé equation, Journal für die Reine und Angewandte Mathematik , to appear, doi: 10.1515/crelle-2014-0082.CrossRefGoogle Scholar
Pillay, A.. Differential fields , Lectures on Algebraic Model Theory, Fields Institute Monographs, vol. 15, American Mathematical Society, Providence, RI, 2002, pp. 1–45.Google Scholar
Pillay, A., Differential algebraic groups and the number of countable differentially closed fields , Model Theory of Fields (Marker, D., Messmer, M., and Pillay, A., editors), ASL Lecture Notes in Logic, vol. 5, A.K. Peters, Ltd., Wellesley, MA, 2006, pp. 111–133.Google Scholar
Rabin, M., Computable algebra, general theory, and theory of computable fields . Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341–360.Google Scholar
Richter, L. J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723–731.Google Scholar
Ritt, J. F., Differential Equations from the Algebraic Standpoint, AMS Colloquium Publications, vol. XIV, American Mathematical Society, New York, 1932.Google Scholar
Sacks, G. E.. Saturated Model Theory, W.A. Benjamin, Reading, 1972.Google Scholar
Shelah, S., Harrington, L., and Makkai, M., A proof of Vaught’s conjecture for ω-stable theories . Israel Journal of Mathematics, vol. 49 (1984), no. 1–3, pp. 259–280.Google Scholar
Slaman, T., Relative to any nonrecursive set . Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2117–2122.Google Scholar
Soare, R. I., Recursively Enumerable Sets and Degrees, Springer-Verlag, New York, 1987.Google Scholar
Soskova, A. A. and Soskov, I. N., A jump inversion theorem for the degree spectra . Journal of Logic and Computation, vol. 19 (2009), no. 1, pp. 199–215.Google Scholar
Thurber, J. J., Every low 2 Boolean algebra has a recursive copy . Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 3859–3866.Google Scholar
Wehner, S., Enumerations, countable structures, and Turing degrees . Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131–2139.Google Scholar