Skip to main content
×
Home
    • Aa
    • Aa

SOME DEFINABLE GALOIS THEORY AND EXAMPLES

  • OMAR LEÓN SÁNCHEZ (a1) and ANAND PILLAY (a2)
Abstract
Abstract

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].

Copyright
References
Hide All
[1] BertrandD. and PillayA., Galois theory, functional Lindemann-Weierstrass, and Manin maps . Pacific Journal of Mathematics, vol. 281 (2016), no. 1, pp. 5182.
[2] BrouetteQ. and PointF.. On Galois groups of strongly normal extensions, preprint, 2015, arXiv:1512.95998v1.
[3] CassidyP. J. and SingerM. F.. Galois theory of parameterized differential equations and linear differential algebraic groups , Differential Equations and Quantum Groups (BertrandD., EnriquezB., MitschiC., SabbahC., and SchaefkeR., editors), IRMA Lectures in Mathematics and Theoretical Physics, vol. 9, EMS Publishing House, Zürich, Switzerland, 2006, pp. 113157.
[4] CrespoT., HajtoZ., and Sowa-AdamusE., Galois correspondence theorem for Picard–Vessiot extensions . Arnold Mathematical Journal, vol. 2 (2016), no. 1, pp. 2127.
[5] DyckerhoffT., The inverse problem of differential Galois theory over the field R(t) , preprint, 2008, arXiv:0802.2897v1.
[6] GilletH., GorchinskiyS., and OvchinnikovA., Parametrized Picard–Vessiot extensions and Atiyah extensions . Advances in Mathematics, vol. 238 (2013), pp. 322411.
[7] HrushovskiE., Unidimensional theories are superstable . Annals of Pure and Applied Logic, vol. 50 (1990), pp. 117138.
[8] HrushovskiE., Computing the Galois group of a linear differential equation , Differential Galois Theory (JaneczkoS., editor), Banach Center Publications, vol. 59, Polish Academy of Sciences, Warszawa, Poland, 2012, pp. 97138.
[9] KamenskyM., Definable groups of automorphisms (Summary of Ph.D. thesis). https://www.math.bgu.ac.il/∼kamenskm/.
[10] KolchinE., Differential Algebra and Algebraic Groups, Academic Press, New York, 1973.
[11] LandesmanP., Generalized differential Galois theory . Transactions of the American Mathematical Society, vol. 360 (2008), pp. 44414495.
[12] León SánchezO., Relative D-groups and differential Galois theory in several derivations . Transactions of the American Mathematical Society, vol. 367 (2015), no. 11, pp. 76137638.
[13] León SánchezO. and NaglooJ., On parameterized differential Galois extensions . Journal of Pure and Applied Algebra, vol. 220 (2016), no. 7, pp. 25492563.
[14] MagidA. R., Differential Galois theory . Notices of the AMS, vol. 46 (1999), no. 9, pp. 10411049.
[15] MarkerD., MessmerM., and PillayA., Model Theory of Fields , Lectures Note in Logic, vol. 5, Association of Symbolic Logic, Diego, CA, USA, 1996.
[16] MedvedevA. and Takloo-BigashR., An invitation to model-theoretic Galois theory . Bulletin of Symbolic Logic, vol. 16 (2010), no. 2, pp. 261269.
[17] McGrailT., The model theory of differential fields with finitely many commuting derivations . Journal of Symbolic Logic, vol. 65 (2000), no. 2, pp. 885913.
[18] PillayA., Differential Galois theory I. Illinois Journal of Mathematics, vol. 42 (1998), no. 4, pp. 678699.
[19] PillayA., Algebraic D-groups and differential Galois theory . Pacific Journal of Mathematics, vol. 216 (2004), pp. 343360.
[20] PillayA., Geometric Stability Theory, Oxford University Press, Oxford, UK, 1996.
[21] PillayA., Around differential Galois theory , Algebraic Theory of Differential Equations (MacCallumM. A. H. and MikhailovA. V., editors), LMS Lecture Notes series, Cambridge University Press, Cambridge, UK, 2009, pp. 232239.
[22] van der PutM and SingerM. F., Galois Theory of Linear Differential Equations , Grundlehren der mathematischen Wissenschaften, vol. 328, Springer, New York, USA, 2003.
[23] PoizatB., Une théorie de Galois imaginaire . Journal of Symbolic Logic, vol. 48 (1983), no. 4, pp. 11511170.
[24] PoizatB., A Course in Model Theory: An Introduction to Contemporary Mathematical Logic, Springer, New York, USA, 2000.
[25] ZilberB. I., Totally categorical theories: Structural properties and non-finite axiomatizability , Model Theory of Algebra and Arithmetic (PacholskiL., WierzejewskiJ., and WilkieA., editors), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, Heidelberg, New York, 1980, pp. 381410.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 30 *
Loading metrics...

Abstract views

Total abstract views: 109 *
Loading metrics...

* Views captured on Cambridge Core between 21st June 2017 - 24th October 2017. This data will be updated every 24 hours.