Skip to Main content Skip to Navigation
Theses

Applications de la théorie de Galois différentielle aux équations différentielles linéaires d'ordre 4

Abstract : For differential linear equations of order 2 and 3, algorithms for exact computation in realistic time exist, based on a preliminary accurate study of groups which may appear as differential Galois groups of these equations. Previous studies of order 4 dealed with particular points of the classification of the groups. In this thesis, we give give optimal bounds for the degree of the minimal polynom of logarithmic derivatives of liouvillian solutions of such equations (common work with D. Boucher and F. Ulmer) then this gives an algorithmic strategy in order to look for the differential Galois group of an equation by knowing its semi-invariants of degree 2 and 4, which has been obtained especially by studying the behaviour of the monomial subgroups of SL(4, C) (which hadn't been done before). We find then more effectively semi-invariants which are products of linear forms too. In chapter 4 of this thesis, we are interested in degree fall of the fourth symmetric power of an equation. More precisely, we show that an order 1 fall implies the existence of at least one semi-invariant for the equation, which gives informations about its group. In case of bigger fall, conditions of finiteness of the group are given by a theorem of M.F. Singer. In chapter 5, we deal with two examples. In the first, we use the algorithmic strategy of chapter 3 in order to find the differential Galois group of an equation whose liouvillian solutions can be computed thanks to a method developped by F. Ulmer. The second example is a concrete resolution of the inverse problem for SO(4, C) thanks to the method given by C. Mitschi and M.F. Singer (equation without liouvillian solutions). Explicit list of the semi-invariants of the monomial subgroups of SL(4, C) is given in annex.
Document type :
Theses
Complete list of metadatas

Cited literature [64 references]  Display  Hide  Download

https://tel.archives-ouvertes.fr/tel-00008234
Contributor : Marie-Annick Guillemer <>
Submitted on : Monday, January 24, 2005 - 3:34:22 PM
Last modification on : Thursday, January 7, 2021 - 4:22:17 PM
Long-term archiving on: : Friday, April 2, 2010 - 9:14:47 PM

Identifiers

  • HAL Id : tel-00008234, version 1

Citation

Philippe Gaillard. Applications de la théorie de Galois différentielle aux équations différentielles linéaires d'ordre 4. Mathématiques [math]. Université Rennes 1, 2004. Français. ⟨tel-00008234⟩

Share

Metrics

Record views

673

Files downloads

1710