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TURING DEGREE SPECTRA OF DIFFERENTIALLY CLOSED FIELDS

  • DAVID MARKER (a1) and RUSSELL MILLER (a2)

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.

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[1] 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. 573599.
[2] Blum, L., Generalized algebraic theories: A model theoretic approach , Ph.D. thesis, Massachusetts Institute of Technology, 1968.
[3] 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. 3761.
[4] 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. 871880.
[5] 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. 545552.
[6] 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. 755777.
[7] Harrington, L., Recursively presentable prime models, this Journal, vol. 39 (1974), no. 2, pp. 305309.
[8] 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. 71113.
[9] Hrushovski, E. and Itai, M., On model complete differential fields . Transactions of the American Mathematical Society, vol. 355 (2003), no. 11, pp. 42674296.
[10] Hrushovski, E. and Sokolović, Z., Minimal subsets of differentially closed fields, preprint from the early 1990s.
[11] 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. 3964.
[12] Knight, J. F., Degrees coded in jumps of orderings, this Journal, vol. 51 (1986), pp. 10341042.
[13] Knight, J. F. and Stob, M., Computable Boolean algebras, this Journal, vol. 65 (2000), no. 4, pp. 16051623.
[14] 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. 121.
[15] 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. 41109.
[16] Miller, R. G., Computable fields and Galois theory , Notices of the AMS, vol. 55 (2008), no. 7, pp. 798807.
[17] Miller, R., Ovchinnikov, A., and Trushin, D., Computing constraint sets for differential fields . Journal of Algebra, vol. 407 (2014), pp. 316357.
[18] Miller, R., Poonen, B., Schoutens, H., and Shlapentokh, A., A computable functor from graphs to fields, submitted for publication.
[19] 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.
[20] 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.
[21] Pillay, A.. Differential fields , Lectures on Algebraic Model Theory, Fields Institute Monographs, vol. 15, American Mathematical Society, Providence, RI, 2002, pp. 145.
[22] 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. 111133.
[23] Rabin, M., Computable algebra, general theory, and theory of computable fields . Transactions of the American Mathematical Society, vol. 95 (1960), pp. 341360.
[24] Richter, L. J., Degrees of structures, this Journal, vol. 46 (1981), pp. 723731.
[25] Ritt, J. F., Differential Equations from the Algebraic Standpoint, AMS Colloquium Publications, vol. XIV, American Mathematical Society, New York, 1932.
[26] Sacks, G. E.. Saturated Model Theory, W.A. Benjamin, Reading, 1972.
[27] 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. 259280.
[28] Slaman, T., Relative to any nonrecursive set . Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21172122.
[29] Soare, R. I., Recursively Enumerable Sets and Degrees, Springer-Verlag, New York, 1987.
[30] 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. 199215.
[31] Thurber, J. J., Every low 2 Boolean algebra has a recursive copy . Proceedings of the American Mathematical Society, vol. 123 (1995), pp. 38593866.
[32] Wehner, S., Enumerations, countable structures, and Turing degrees . Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 21312139.
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  • ISSN: 0022-4812
  • EISSN: 1943-5886
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