Displaying similar documents to “On the envelope of a vector field”

Reduction of differential equations

Krystyna Skórnik, Joseph Wloka (2000)

Banach Center Publications

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Let (F,D) be a differential field with the subfield of constants C (c ∈ C iff Dc=0). We consider linear differential equations (1) L y = D n y + a n - 1 D n - 1 y + . . . + a 0 y = 0 , where a 0 , . . . , a n ∈ F , and the solution y is in F or in some extension E of F (E ⊇ F). There always exists a (minimal, unique) extension E of F, where Ly=0 has a full system y 1 , . . . , y n of linearly independent (over C) solutions; it is called the Picard-Vessiot extension of F E = PV(F,Ly=0). The Galois group G(E|F) of an extension field E ⊇ F consists of all differential automorphisms...

Differential equations and algebraic transcendents: french efforts at the creation of a Galois theory of differential equations 1880–1910

Tom Archibald (2011)

Revue d'histoire des mathématiques

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A “Galois theory” of differential equations was first proposed by Émile Picard in 1883. Picard, then a young mathematician in the course of making his name, sought an analogue to Galois’s theory of polynomial equations for linear differential equations with rational coefficients. His main results were limited by unnecessary hypotheses, as was shown in 1892 by his student Ernest Vessiot, who both improved Picard’s results and altered his approach, leading Picard to assert that his lay...

On the inverse problem of Galois theory.

Núria Vila (1992)

Publicacions Matemàtiques

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The problem of the construction of number fields with Galois group over Q a given finite groups has made considerable progress in the recent years. The aim of this paper is to survey the current state of this problem, giving the most significant methods developed in connection with it.

On the problem of Baer and Kolchin in the Picard-Vessiot theory

Beata Kocel-Cynk, Elżbieta Sowa (2011)

Banach Center Publications

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We present the history of the development of Picard-Vessiot theory for linear ordinary differential equations. We are especially concerned with the condition of not adding new constants, pointed out by R. Baer. We comment on Kolchin's condition of algebraic closedness of the subfield of constants of the given differential field over which the equation is defined. Some new results concerning existence of a Picard-Vessiot extension for a homogeneous linear ordinary differential equation...

Remarks on the intrinsic inverse problem

Daniel Bertrand (2002)

Banach Center Publications

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The intrinsic differential Galois group is a twisted form of the standard differential Galois group, defined over the base differential field. We exhibit several constraints for the inverse problem of differential Galois theory to have a solution in this intrinsic setting, and show by explicit computations that they are sufficient in a (very) special situation.

Realizability and automatic realizability of Galois groups of order 32

Helen Grundman, Tara Smith (2010)

Open Mathematics

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This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.