Morales-Ramis theorems via Malgrange pseudogroup
type: Item
instance of: scholarly article
title: Morales-Ramis theorems via Malgrange pseudogroup
zbMATH Open document ID: 1232.12005
author: Guy Casale
publication date: 2010-04-06T00:00:00Z
full work available at URL: https://eudml.org/doc/10465
review text: In this paper the authors prove the following results in algebraic setting for integrability by quadratures. Definition: Let \(E(t,y,y',\dots,y^{(n)})\in C[t,y,y',\dots,y^{(n)}]\) be a differential equation of order \(n\) given by an irreducible polynomial. A Liouvillian solution is a solution \(f\) in a differential extension \(K_N\) of \(K_0=C(t)\) built by successive elementary extensions \(K_{i-1}\subset K_i\), \(1\leq i \leq N\), of the form \(K_i=K_{i-1}(u_i)\) with \(u_i\) algebraic over \(K_{i-1}\) or \(u'_i\in K_{i-1}\) or \(\frac{u'_i}{u_i}\in K_{i-1}\). The equation \(E\) is said to be integrable by quadrature if there is a Liouvillian solution \(f\) with \(\text{transc.deg.}\,C(t,y,y',\dots,y^{(n)})/ C(t) =n \) Theorem: If a rational differential equation is integrable by quadratures then the Galois group of its variational equation of order \(q\) along an algebraic solution is virtually solvable. If \(N\) elementary extensions are needed to build the general solution, then the \(N\)th derived Lie algebra of the Galois group is zero.
Mathematics Subject Classification ID: 12H05
Mathematics Subject Classification ID: 37J30
Mathematics Subject Classification ID: 34M15
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 58H05
zbMATH DE Number: 5689400
zbMATH Keywords: differential Galois theory
zbMATH Keywords: variational equation
zbMATH Keywords: integrability
MaRDI profile type: Publication
OpenAlex ID: W4299301812
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DOI: 10.5802/AIF.2501
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5689400