Hostname: page-component-8448b6f56d-tj2md Total loading time: 0 Render date: 2024-04-19T22:33:14.090Z Has data issue: false hasContentIssue false
  • English
  • Français

Iterative differential Galois theory in positive characteristic: A model theoretic approach

Published online by Cambridge University Press:  12 March 2014

Javier Moreno
Javier Moreno*
Affiliation:
Institut Camille Jordan, Université Claude Bernard Lyon1, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne Cedex, France, E-mail: moreno@math.univ-lyonl.fr, URL: http://math.univ-lyonl.fr/~moreno/
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper introduces a natural extension of Kolchin's differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard–Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a G-primitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 1 , March 2011 , pp. 125 - 142
Copyright
Copyright © Association for Symbolic Logic 2011

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]André, Y., Différentielles non-commutatives et théorie de galois différentielle ou aux différences, Annales Scientifiques de l'Ecole Normale Supérieure, vol. 34 (2001), no. 5, pp. 685–739.CrossRefGoogle Scholar
[2]Benoist, F., Théorie des modèles des corps munis d'une dérivation de Hasse, Ph.D. thesis, Équipe de Logique Mathématique, Université Paris 7 – Denis Diderot, 2005, Available online at http://tel.archives-ouvertes.fr/tel-00134889.Google Scholar
[3]Delon, F., Separably closed fields, Model theory and algebraic geometry: An introduction to E. Hrushovski's proof of the geometric Mordell–Lang conjecture (Bouscaren, E., editor), Lecture Notes in Mathematics, vol. 1696, Springer, 1999, pp. 143–176.CrossRefGoogle Scholar
[4]van den Dries, L. P. D., Weil's group chunk theorem: A topological setting, Illinois Journal of Mathematics, vol. 34 (1990), no. 1, pp. 127–139.CrossRefGoogle Scholar
[5]Hrushovski, E., Unidimensional theories are superstable, Annals of Pure and Applied Logic, vol. 50 (1990), pp. 117–138.CrossRefGoogle Scholar
[6]Hrushovski, E., Computing the Galois group of a linear differential equation, Differential Galois theory (Bedlewo, 2001), vol. 58, Polish Academy of Sciences, Banach Center Publication, Warsaw, 2002, pp. 97–138.CrossRefGoogle Scholar
[7]Kolchin, E. R., Differential algebra and algebraic groups, Academic Press, 1973.Google Scholar
[8]Kovacic, J., The differential Galois theory of strongly normal extensions, Transactions of the American Mathematical Society, vol. 355 (2003), no. 11, pp. 4475–4522.CrossRefGoogle Scholar
[9]Magid, A., Lectures on differential Galois theory, University Lecture Series, vol. 7, American Mathematical Society, 1994.Google Scholar
[10]Matzat, H., Differential Galois theory in positive characteristic, notes written by Julia Hartmann, preprint, 2001.Google Scholar
[11]Messmer, M. and Wood, C., Separably closedfields with higher derivations, this Journal, vol. 60 (1995), no. 3, pp. 898–910.Google Scholar
[12]Moosa, R., Pillay, A., and Scanlon, T., Differential arcs and regular types in differential fields, Journal für die Reine und Angewandte Mathematik, vol. 620 (2008), pp. 35–54.Google Scholar
[13]Okugawa, K., Basic properties of differential fields of arbitrary characteristic and the Picard–Vessiot theory, Journal of Mathematics of Kyoto University, vol. 2 (1963), no. 3, pp. 294–322.Google Scholar
[14]Okugawa, K., Differential algebra of nonzero characteristic, Lectures in Mathematics 16, Kinokuniya Company Ltd., Tokyo, 1987.Google Scholar
[15]Pillay, A., Differential Galois theory II, Annals of Pure and Applied Logic, vol. 88 (1997), pp. 181–191.CrossRefGoogle Scholar
[16]Pillay, A., Differential Galois theory I, Illinois Journal of Mathematics, vol. 42 (1998), no. 4, pp. 678–699.CrossRefGoogle Scholar
[17]Pillay, A., Two remarks on differential fields, Model theory and applications, Quaderni di Matematica, vol. 11, 2002.Google Scholar
[18]Pillay, A., Algebraic D-groups and differential Galois theory, Pacific Journal of Mathematics, vol. 216 (2004), pp. 343–360.CrossRefGoogle Scholar
[19]Pillay, A. and Marker, D., Differential Galois theory III: Some inverse problems, Illinois Journal of Mathematics, vol. 3 (1997), pp. 453–461.Google Scholar
[20]Pillay, A. and Sokolović, Ž., Superstable differential fields, this Journal, vol. 56 (1992), no. 1, pp. 97–108.Google Scholar
[21]Poizat, B., Une théorie de Galois imaginaire, this Journal, vol. 48 (1983), no. 4, pp. 1151–1170.Google Scholar
[22]Poizat, B., Stable groups, American Mathematical Society, 2001.CrossRefGoogle Scholar
[23]van der Put, M. and Singer, M., Galois theory of linear differential equations, Springer, 2003.CrossRefGoogle Scholar
[24]Shikishima-Tsuji, K., Galois theory of differential fields of positive characteristic, Pacific Journal of Mathematics, vol. 138 (1989), no. 1, pp. 151–168.CrossRefGoogle Scholar
[25]Umemura, H., Galois theory and Painlevé equations, Séminaires et Congrès, vol. 14 (2006), pp. 299–339.Google Scholar
[26]Ziegler, M., Separably closed fields with Hasse derivations, this Journal, vol. 68 (2003), no. 1, pp. 311–318.Google Scholar
[27]Zilber, B., Totally categorical theories: structural properties and non-finite axiomatizability, Model theory of algebra and arithmetic (Pacholski, L.et al., editors), Springer, 1980, pp. 381–410.CrossRefGoogle Scholar