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SOME DEFINABLE GALOIS THEORY AND EXAMPLES

Published online by Cambridge University Press:  21 June 2017

OMAR LEÓN SÁNCHEZ  and
ANAND PILLAY
OMAR LEÓN SÁNCHEZ
Affiliation:
SCHOOL OF MATHEMATICS UNIVERSITY OF MANCHESTER OXFORD ROAD MANCHESTER M13 9PL, UKE-mail: omar.sanchez@manchester.ac.uk
ANAND PILLAY
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF NOTRE DAME 255 HURLEY, NOTRE DAME IN 46556, USAE-mail: apillay@nd.edu
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Abstract

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We make explicit certain results around the Galois correspondence in the context of definable automorphism groups, and point out the relation to some recent papers dealing with the Galois theory of algebraic differential equations when the constants are not “closed” in suitable senses. We also improve the definitions and results on generalized strongly normal extensions from [Pillay, “Differential Galois theory I”, Illinois Journal of Mathematics, 42(4), 1998], using this to give a restatement of a conjecture on almost semiabelian δ-groups from [Bertrand and Pillay, “Galois theory, functional Lindemann–Weierstrass, and Manin maps”, Pacific Journal of Mathematics, 281(1), 2016].

Keywords

03C6012H05model theorydifferential fieldsstrongly normal extensions
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 23 , Issue 2 , June 2017 , pp. 145 - 159
Copyright
Copyright © The Association for Symbolic Logic 2017 

References

REFERENCES

Bertrand, D. and Pillay, A., Galois theory, functional Lindemann-Weierstrass, and Manin maps . Pacific Journal of Mathematics, vol. 281 (2016), no. 1, pp. 51–82.CrossRefGoogle Scholar
Brouette, Q. and Point, F.. On Galois groups of strongly normal extensions, preprint, 2015, arXiv:1512.95998v1.Google Scholar
Cassidy, P. J. and Singer, M. F.. Galois theory of parameterized differential equations and linear differential algebraic groups , Differential Equations and Quantum Groups (Bertrand, D., Enriquez, B., Mitschi, C., Sabbah, C., and Schaefke, R., editors), IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, EMS Publishing House, Zürich, Switzerland, 2006, pp. 113–157.CrossRefGoogle Scholar
Crespo, T., Hajto, Z., and Sowa-Adamus, E., Galois correspondence theorem for Picard–Vessiot extensions . Arnold Mathematical Journal, vol. 2 (2016), no. 1, pp. 21–27.CrossRefGoogle Scholar
Dyckerhoff, T., The inverse problem of differential Galois theory over the field R(t) , preprint, 2008, arXiv:0802.2897v1.Google Scholar
Gillet, H., Gorchinskiy, S., and Ovchinnikov, A., Parametrized Picard–Vessiot extensions and Atiyah extensions . Advances in Mathematics, vol. 238 (2013), pp. 322–411.CrossRefGoogle Scholar
Hrushovski, E., Unidimensional theories are superstable . Annals of Pure and Applied Logic, vol. 50 (1990), pp. 117–138.CrossRefGoogle Scholar
Hrushovski, E., Computing the Galois group of a linear differential equation , Differential Galois Theory (Janeczko, S., editor), Banach Center Publications, vol. 59, Polish Academy of Sciences, Warszawa, Poland, 2012, pp. 97–138.Google Scholar
Kamensky, M., Definable groups of automorphisms (Summary of Ph.D. thesis). https://www.math.bgu.ac.il/∼kamenskm/.Google Scholar
Kolchin, E., Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.Google Scholar
Landesman, P., Generalized differential Galois theory . Transactions of the American Mathematical Society, vol. 360 (2008), pp. 4441–4495.CrossRefGoogle Scholar
León Sánchez, O., Relative D-groups and differential Galois theory in several derivations . Transactions of the American Mathematical Society, vol. 367 (2015), no. 11, pp. 7613–7638.CrossRefGoogle Scholar
León Sánchez, O. and Nagloo, J., On parameterized differential Galois extensions . Journal of Pure and Applied Algebra, vol. 220 (2016), no. 7, pp. 2549–2563.CrossRefGoogle Scholar
Magid, A. R., Differential Galois theory . Notices of the AMS, vol. 46 (1999), no. 9, pp. 1041–1049.Google Scholar
Marker, D., Messmer, M., and Pillay, A., Model Theory of Fields , Lectures Note in Logic, vol. 5, Association of Symbolic Logic, Diego, CA, USA, 1996.Google Scholar
Medvedev, A. and Takloo-Bigash, R., An invitation to model-theoretic Galois theory . Bulletin of Symbolic Logic, vol. 16 (2010), no. 2, pp. 261–269.CrossRefGoogle Scholar
McGrail, T., The model theory of differential fields with finitely many commuting derivations . Journal of Symbolic Logic, vol. 65 (2000), no. 2, pp. 885–913.CrossRefGoogle Scholar
Pillay, A., Differential Galois theory I. Illinois Journal of Mathematics, vol. 42 (1998), no. 4, pp. 678–699.Google Scholar
Pillay, A., Algebraic D-groups and differential Galois theory . Pacific Journal of Mathematics, vol. 216 (2004), pp. 343–360.CrossRefGoogle Scholar
Pillay, A., Geometric Stability Theory, Oxford University Press, Oxford, UK, 1996.CrossRefGoogle Scholar
Pillay, A., Around differential Galois theory , Algebraic Theory of Differential Equations (MacCallum, M. A. H. and Mikhailov, A. V., editors), LMS Lecture Notes series, Cambridge University Press, Cambridge, UK, 2009, pp. 232–239.Google Scholar
van der Put, M and Singer, M. F., Galois Theory of Linear Differential Equations , Grundlehren der mathematischen Wissenschaften, vol. 328, Springer, New York, USA, 2003.Google Scholar
Poizat, B., Une théorie de Galois imaginaire . Journal of Symbolic Logic, vol. 48 (1983), no. 4, pp. 1151–1170.CrossRefGoogle Scholar
Poizat, B., A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, New York, USA, 2000.CrossRefGoogle Scholar
Zilber, B. I., Totally categorical theories: Structural properties and non-finite axiomatizability , Model Theory of Algebra and Arithmetic (Pacholski, L., Wierzejewski, J., and Wilkie, A., editors), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, New York, 1980, pp. 381–410.CrossRefGoogle Scholar