Q1613959
type: Item
instance of: scholarly article
title: Factorisation d'opérateurs différentiels à coefficients dans une extension liouvillienne d'un corps valué. (Factorization of differential operators with coefficients in a Liouvillian extension of a valued field)
zbMATH Open document ID: 1017.12006
author: Magali Bouffet
publication date: 2002-09-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/115992
review text: Let \(\mathbb C\) be the field of complex numbers, \({\mathbb C}((z))\) be the field of formal Laurent series in \(z\) with coefficients in \(\mathbb C\) endowed with the derivation operator \(\delta = -z^2d/dz\). By \(L\) we denote the field \({\mathbb C}((z))((e^{1/z}))\). The goal of the author is the study of linear differential equations with coefficients in \(L\) in order to construct a differential Galois theory over \(L\). In particular, the solutions of linear differential equations with coefficients in \(L\) should be determined or, in other words, differential operators with coefficients in \(L\) should be factorized. This class of fields is of interest, because the coefficients of the differential equations admit essential singularities and not only poles. The following theorem is proven. Theorem. Consider the field \(K={\mathbb C}((z))\) endowed with the \(z\)-adic valuation and the derivation operator \(\delta=ad/dz\), where \(a\in{\mathbb C}((z))\). Let \(\mu_1,\dots,\mu_p\) be elements of \({\mathbb C}((z))\) such that \[ X_1=e^{\mu_1},\dots,X_n=e^{\mu_n},X_{n+1}=\log(\mu_{n+1}),\dots,X_p=\log(\mu_p) \] are algebraically independent over \({\mathbb C}((z))\). Let \(K\subset L=K(X_1,\dots,X_p)\). By \(F_0\) we denote the algebraic closure of \(K\) and, for any \(i\), \(1\leq i\leq p\), we define \(F_i=\bigcup_{n\geq 1}F_{i-1}((X_i^{1/n}))\). If \(P\in L[\delta]\) is a nonzero differential operator, then \(P\) can be factorized into a product of first-order operators in the ring \(F_p[\delta]\).
Mathematics Subject Classification ID: 12H20
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 34M50
zbMATH DE Number: 1794812
zbMATH Keywords: differential field
zbMATH Keywords: valued field
zbMATH Keywords: Newton polygon
zbMATH Keywords: Hensel lemma
zbMATH Keywords: factorization
zbMATH Keywords: discrete valuations
zbMATH Keywords: linear differential equations
zbMATH Keywords: Liouvillian extensions
reviewed by: E. V. Pankrat'ev
MaRDI profile type: Publication
OpenAlex ID: W2478660843
cites work: Q4071850
cites work: Un lemme de Hensel pour les opérateurs différentiels
cites work: Q3882586
cites work: Jordan decomposition for a class of singular differential operators
cites work: Differential equations with polynomial coefficients
cites work: Lemmes de Hensel pour les opérateurs différentiels. Application à la reduction formelle des équations différentielles
cites work: Q5796078
cites work: Q4248267
DOI: 10.5802/AIF.1899
published in: Annales de l’institut Fourier
label: Factorisation d'opérateurs différentiels à coefficients dans une extension liouvillienne d'un corps valué. (Factorization of differential operators with coefficients in a Liouvillian extension of a valued field)
description: scientific article; zbMATH DE number 1794812