Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-29T01:26:43.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential forms in the model theory of differential fields

Published online by Cambridge University Press:  12 March 2014

David Pierce
David Pierce*
Affiliation:
Mathematics Department, Middle East Technical University, Ankara 06531, Turkey, E-mail: [email protected], URL: http://www.math.metu.edu.tr/~dpierce/
Get access Rights & Permissions [Opens in a new window]

Abstract

Fields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 68 , Issue 3 , September 2003 , pp. 923 - 945
Copyright
Copyright © Association for Symbolic Logic 2003

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Barwise, Jon (editor), with the cooperation of Keisler, H. J., Kunen, K., Moschovakis, Y. N. and Troelstra, A. S., Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977.Google Scholar
[2] Chatzidakis, Zoé and Hrushovski, Ehud, Model theory of difference fields, Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071.Google Scholar
[3] Chern, Shiing-Shen and Chevalley, Claude, Obituary: Elie Carian and his mathematical work, Bulletin of the American Mathematical Society, vol. 58 (1952), pp. 217–250.Google Scholar
[4] Hodges, Wilfrid, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993.CrossRefGoogle Scholar
[5] Jacobson, Nathan, Basic algebra. II, W. H. Freeman and Co., San Francisco, California, 1980.Google Scholar
[6] Johnson, Joseph, Reinhart, Georg M., and Rubel, Lee A., Some counterexamples to separation of variables, Journal of Differential Equations, vol. 121 (1995), no. 1, pp. 42–66.Google Scholar
[7] Lang, Serge, Introduction to algebraic geometry, Interscience Publishers, Inc., New York-London, 1958.Google Scholar
[8] Macintyre, Angus, Generic automorphisms of fields, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 165–180, Joint AILA-KGS Model Theory Meeting (Florence, 1995).Google Scholar
[9] McGrail, Tracey, The model theory of differential fields with finitely many commuting derivations, this Journal, vol. 65 (2000), no. 2, pp. 885–913.Google Scholar
[10] Pierce, David and Pillay, Anand, A note on the axioms for differentially closed fields of characteristic zero, Journal of Algebra, vol. 204 (1998), no. 1, pp. 108–115.Google Scholar
[11] Pillay, Anand, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press Oxford University Press, New York, 1996, Oxford Science Publications.Google Scholar
[12] Pillay, Anand, Differential fields, Lectures on algebraic model theory, Fields Institute Monographs, vol. 15, American Mathematical Society, Providence, Rhode Island, 2002, pp. 1–45.Google Scholar
[13] Robinson, A., Solution of a problem of Tarski, Fundamenta Mathematicae, vol. 47 (1959), pp. 179–204.Google Scholar
[14] Sharpe, R. W., Differential geometry, Cartan's generalization of Klein's Erlangen program, Graduate Texts in Mathematics, vol. 166, Springer-Verlag, New York, 1997, With a foreword by S. S. Chern.Google Scholar
[15] Spivak, Michael, A comprehensive introduction to differential geometry. Volume I, second ed., Publish or Perish Inc., Wilmington, Delaware, 1979.Google Scholar
[16] Yaffe, Yoav, Model completion of Lie differential fields, Annals of Pure and Applied Logic, vol. 107 (2001), no. 1–3, pp. 49–86.CrossRefGoogle Scholar