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A short introduction to differential Galois theory

  • II-Algebraic System Theory
  • Conference paper
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New Trends in Nonlinear Control Theory

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 122))

Abstract

In the first part of these notes we will give a brief description of the “classical” differential Galois theory (for more details see [Pi], [Ve], [Kap], [Ko1], [Be1], [Sin1]). One problem with the classical theory is the difficulty of explicit calculations : from the birth of our subject (late 19th century) until to very recent work ([Kat4], [KP], [B.B.H], [B.H], [Ra5], [Ra8], [DM]) the only explicit computations we know are for Airy equation [Kap], and Bessel equations [Ko2] (and in fact Airy equation can be reduced to a special case of Bessel equation [AS]...), but for evident situations. In the second part of our paper we will give a new description of the differential Galois theory (when the “field of constants” is the complex field C) in relation with recent progress on the problem of classification of analytic differential equations in the complex domain up to analytic transformations ([Si], [Ma1], [Ma2], [BJL1], [BJL2], [J], [BV1], [BV2], [Ra7], [Ra2], for the linear case, and [M.R.1], [M.R2], [MR3], [E3] for the non linear case), and with a new theory of asymptotics ([Ra1], [Ra2], [Ra4], [Ra7], [RS1], [MR1], [E1], [E2] [E3], [E4], [EMMR1], [EMMR2], [MR4]). Using this description it is in particular possible to get a method of computation for the case of Meijer G-functions, that is for more or less all the cases of special functions solutions of linear differential equations [Er] (cf. [DM], [Ra8], [BH]).

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  1. Université de Strasbourg 1, Strasbourg, France

    J. P. Ramis

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  1. J. P. Ramis
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J. Descusse Michel Fliess A. Isidori D. Leborgne

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© 1989 Springer-Verlag

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Ramis, J.P. (1989). A short introduction to differential Galois theory. In: Descusse, J., Fliess, M., Isidori, A., Leborgne, D. (eds) New Trends in Nonlinear Control Theory. Lecture Notes in Control and Information Sciences, vol 122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0043024

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  • DOI: https://doi.org/10.1007/BFb0043024

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