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Book Review

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Book Information:

Authors: Marius van der Put and Michael Singer
Title: Galois theory of linear differential equations
Additional book information: Springer-Verlag, Berlin, 2003, xviii+438 pp., ISBN 3-540-44228-6, $99.00$

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

Yves André, Différentielles non commutatives et théorie de Galois différentielle ou aux différences, Ann. Sci. École Norm. Sup. (4) 34 (2001), no. 5, 685–739 (French, with English and French summaries). MR 1862024, DOI 10.1016/S0012-9593(01)01074-6

D. Bertrand, Théorie de Galois différentielle, Cours de DEA, Université de Paris VI, 1986.

Daniel Bertrand, Groupes algébriques et équations différentielles linéaires, Astérisque 206 (1992), Exp. No. 750, 4, 183–204 (French, with French summary). Séminaire Bourbaki, Vol. 1991/92. MR 1206068

D. Bertrand, Review of the book Lectures on Differential Galois Theory by A. Magid, Bulletin of the Amer. Math. Soc. (N.S.) 33 (1996) 289-294.

A. A. Bolibrukh, The Riemann-Hilbert problem, Uspekhi Mat. Nauk 45 (1990), no. 2(272), 3–47, 240 (Russian); English transl., Russian Math. Surveys 45 (1990), no. 2, 1–58. MR 1069347, DOI 10.1070/RM1990v045n02ABEH002350

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

Alexandru Buium, Differential function fields and moduli of algebraic varieties, Lecture Notes in Mathematics, vol. 1226, Springer-Verlag, Berlin, 1986. MR 874111, DOI 10.1007/BFb0101622

P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14

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

Irving Kaplansky, An introduction to differential algebra, 2nd ed., Publications de l’Institut de Mathématique de l’Université de Nancago, No. V, Hermann, Paris, 1976. MR 0460303

Nicholas M. Katz, A conjecture in the arithmetic theory of differential equations, Bull. Soc. Math. France 110 (1982), no. 2, 203–239 (English, with French summary). MR 667751

Nicholas M. Katz, On the calculation of some differential Galois groups, Invent. Math. 87 (1987), no. 1, 13–61. MR 862711, DOI 10.1007/BF01389152

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

Michèle Loday-Richaud, Stokes phenomenon, multisummability and differential Galois groups, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 3, 849–906 (English, with English and French summaries). MR 1303886

Bernard Malgrange, Le groupoïde de Galois d’un feuilletage, Essays on geometry and related topics, Vol. 1, 2, Monogr. Enseign. Math., vol. 38, Enseignement Math., Geneva, 2001, pp. 465–501 (French). MR 1929336

B. Malgrange, On nonlinear differential Galois theory, Chinese Ann. Math. Ser. B 23 (2002), no. 2, 219–226. Dedicated to the memory of Jacques-Louis Lions. MR 1924138, DOI 10.1142/S0252959902000213

J.-P. Ramis and J. Martinet, Théorie de Galois différentielle et resommation, Computer algebra and differential equations, Comput. Math. Appl., Academic Press, London, 1990, pp. 117–214 (French). MR 1038060

Juan J. Morales Ruiz, Differential Galois theory and non-integrability of Hamiltonian systems, Progress in Mathematics, vol. 179, Birkhäuser Verlag, Basel, 1999. MR 1713573, DOI 10.1007/978-3-0348-8718-2

20.

E. Picard, Sur les groupes de transformation des équations différentielles linéaires, C.R. Acad. Sci. Paris 96 (1883) 1131-1134.

21.

E. Picard, Sur équations différentielles et les groupes algébriques des transformations, Ann. Fac. Sci. Univ. de Toulouse (1) 1 (1887), A1-A15.

22.

E. Picard, Traité d'Analyse, Tome III, Gauthiers-Villars, Paris, 1928.

Josip Plemelj, Problems in the sense of Riemann and Klein, Interscience Tracts in Pure and Applied Mathematics, No. 16, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1964. Edited and translated by J. R. M. Radok. MR 0174815

Marius van der Put, Recent work on differential Galois theory, Astérisque 252 (1998), Exp. No. 849, 5, 341–367. Séminaire Bourbaki. Vol. 1997/98. MR 1685624

Ellis Kolchin, Selected works of Ellis Kolchin with commentary, American Mathematical Society, Providence, RI, 1999. Commentaries by Armand Borel, Michael F. Singer, Bruno Poizat, Alexandru Buium and Phyllis J. Cassidy; Edited and with a preface by Hyman Bass, Buium and Cassidy. MR 1677530

Hiroshi Umemura, Differential Galois theory of infinite dimension, Nagoya Math. J. 144 (1996), 59–135. MR 1425592, DOI 10.1017/S0027763000006024

V. S. Varadarajan, Linear meromorphic differential equations: a modern point of view, Bull. Amer. Math. Soc. (N.S.) 33 (1996), no. 1, 1–42. MR 1339809, DOI 10.1090/S0273-0979-96-00624-6

28.

M.E. Vessiot, Sur l'intégration des équations différentielles linéaires, Ann. Sci. École Norm. Sup. (3) 9 (1892) 197-280.

Wolfgang Wasow, Asymptotic expansions for ordinary differential equations, Dover Publications, Inc., New York, 1987. Reprint of the 1976 edition. MR 919406

Review Information:

Reviewer: Juan J. Morales-Ruiz
Affiliation: University Politecnica de Catalunya
Email: juan.morales-ruiz@upc.es