In
differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A n ...
, Picard–Vessiot theory is the study of the
differential field extension generated by the solutions of a
linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form
:a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
, using the
differential Galois group
In mathematics, differential Galois theory studies the Galois groups of differential equations.
Overview
Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential field ...
of the field extension. A major goal is to describe when the differential equation can be solved by quadratures in terms of properties of the differential Galois group. The theory was initiated by
Émile Picard
Charles Émile Picard (; 24 July 1856 – 11 December 1941) was a French mathematician. He was elected the fifteenth member to occupy seat 1 of the Académie française in 1924.
Life
He was born in Paris on 24 July 1856 and educated there at t ...
and
Ernest Vessiot from about 1883 to 1904.
and give detailed accounts of Picard–Vessiot theory.
History
The history of Picard–Vessiot theory is discussed by .
Picard–Vessiot theory was developed by Picard between 1883 and 1898 and by Vessiot from 1892 to 1904 (summarized in and ). The main result of their theory says very roughly that a linear differential equation can be solved by quadratures if and only if its differential Galois group is connected and
solvable. Unfortunately it is hard to tell exactly what they proved as the concept of being "solvable by quadratures" is not defined precisely or used consistently in their papers. gave precise definitions of the necessary concepts and proved a rigorous version of this theorem.
extended Picard–Vessiot theory to partial differential fields (with several commuting derivations).
described an algorithm for deciding whether second order homogeneous linear equations can be solved by quadratures, known as Kovacic's algorithm.
Picard–Vessiot extensions and rings
An extension ''F'' ⊆ ''K'' of differential fields is called a Picard–Vessiot extension if all constants are in ''F'' and ''K'' can be generated by adjoining the solutions of a homogeneous linear ordinary differential polynomial.
A Picard–Vessiot ring ''R'' over the differential field ''F'' is a differential ring over ''F'' that is simple (no differential ideals other than 0 and ''R'') and generated as a ''k''-algebra by the coefficients of ''A'' and 1/det(''A''), where ''A'' is an invertible matrix over ''F'' such that ''B'' = ''A''′/''A'' has coefficients in ''F''. (So ''A'' is a fundamental matrix for the differential equation ''y''′ = ''By''.)
Liouvillian extensions
An extension ''F'' ⊆ ''K'' of differential fields is called Liouvillian if all constants are in ''F'', and ''K'' can be generated by adjoining a finite number of integrals, exponential of integrals, and algebraic functions. Here, an integral of an element ''a'' is defined to be any solution of ''y''′ = ''a'', and an exponential of an integral of ''a'' is defined to be any solution of ''y''′ = ''ay''.
A Picard–Vessiot extension is Liouvillian if and only if the
identity component
In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element.
In point set topology, the identity compo ...
of its differential Galois group is solvable (, ). More precisely, extensions by algebraic functions correspond to finite differential Galois groups, extensions by integrals correspond to subquotients of the differential Galois group that are 1-dimensional and unipotent, and extensions by exponentials of integrals correspond to subquotients of the differential Galois group that are 1-dimensional and reductive (tori).
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{{DEFAULTSORT:Picard-Vessiot theory
Differential algebra