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Book Review

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Book Information:

Author: Andy R. Magid
Title: Lectures on differential Galois theory
Additional book information: University Lecture Series, vol. 7, Amer. Math. Soc., Providence, RI, 1994, xiii+105 pp., ISBN 0-8218-7004-1, $35.00$

Yves André, Quatre descriptions des groupes de Galois différentiels, Séminaire d’algèbre Paul Dubreil et Marie-Paule Malliavin (Paris, 1986) Lecture Notes in Math., vol. 1296, Springer, Berlin, 1987, pp. 28–41 (French). MR 932051, DOI 10.1007/BFb0078522

B. L. J. Braaksma and M. van der Put, Analytic and algebraic aspects of complex analytic differential equations, preprint, Groningen, 1994.

Frits Beukers, W. Dale Brownawell, and Gert Heckman, Siegel normality, Ann. of Math. (2) 127 (1988), no. 2, 279–308. MR 932298, DOI 10.2307/2007054

Lawrence Breen, Tannakian categories, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 337–376. MR 1265536

P. Deligne, Catégories tannakiennes, The Grothendieck Festschrift, Vol. II, Progr. Math., vol. 87, Birkhäuser Boston, Boston, MA, 1990, pp. 111–195 (French). MR 1106898

Abdelmajid Fahim, Extensions galoisiennes d’algèbres différentielles, C. R. Acad. Sci. Paris Sér. I Math. 314 (1992), no. 1, 1–4 (French, with English summary). MR 1149627, DOI 10.2140/pjm.1997.180.7

E. Galois, Mémoire sur les conditions de résolubilité des équations par radicaux (R. Bourgne and J.-P. Azra, eds.), Gauthiers-Villars, 1962. [The quotation reads as follows: ``que toute fonction des racines, déterminable rationnellement, soit invariable par ces substitutions.'']

Irving Kaplansky, An introduction to differential algebra, Publ. Inst. Math. Univ. Nancago, No. 5, Hermann, Paris, 1957. MR 0093654

N. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), 203--239.

Nicholas M. Katz, Exponential sums and differential equations, Annals of Mathematics Studies, vol. 124, Princeton University Press, Princeton, NJ, 1990. MR 1081536, DOI 10.1515/9781400882434

Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336

E. R. Kolchin, Differential algebra and algebraic groups, Pure and Applied Mathematics, Vol. 54, Academic Press, New York-London, 1973. MR 0568864

J. Kovacic, On the inverse problem in the Galois theory of differential fields. II, Ann. of Math. (2) 93 (1971), 269–284. MR 285514, DOI 10.2307/1970775

Michio Kuga, Galois’ dream: group theory and differential equations, Birkhäuser Boston, Inc., Boston, MA, 1993. Translated from the 1968 Japanese original by Susan Addington and Motohico Mulase. MR 1199112, DOI 10.1007/978-1-4612-0329-2

15.

C. Mitschi and M. Singer, Connected linear groups as differential Galois groups, to appear in J. Algebra.

Jean-Pierre Ramis, Phénomène de Stokes et filtration Gevrey sur le groupe de Picard-Vessiot, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 5, 165–167 (French, with English summary). MR 801953

Jean-Pierre Serre, Gèbres, Enseign. Math. (2) 39 (1993), no. 1-2, 33–85 (French). MR 1225256

Michael F. Singer and Felix Ulmer, Galois groups of second and third order linear differential equations, J. Symbolic Comput. 16 (1993), no. 1, 9–36. MR 1237348, DOI 10.1006/jsco.1993.1032

Review Information:

Reviewer: D. Bertrand
Affiliation: Institut de Mathématiques, Université de Paris VI
Email: bertrand@mathp6.jussieu.fr