Galois Theory of Linear Differential Equations

Put, Marius van der; Singer, Michael F.

doi:10.1007/978-3-642-55750-7

Cited by 680 publications

(956 citation statements)

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“…Therefore (CD( [ ]), ) is recovered from the usual algebra of differential polynomials { } over (see [4,5] for instance), that is, the free commutative -algebra [ × N] with the -derivation ( , ) = ( , + 1) for all ∈ , ≥ 0. Now, if ⟨ ⟩ = T( ) is the free -algebra over , then (D( ⟨ ⟩), ) is the (not so wellknown, see [6] however) noncommutative counterpart of { }, that is, the free -algebra ⟨ × N⟩ with derivation…”

Section: Examplementioning

confidence: 99%

Wronskian Envelope of a Lie Algebra

Poinsot

2013

Algebra

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The famous Poincaré-Birkhoff-Witt theorem states that a Lie algebra, free as a module, embeds into its associative envelope—its universal enveloping algebra—as a sub-Lie algebra for the usual commutator Lie bracket. However, there is another functorial way—less known—to associate a Lie algebra to an associative algebra and inversely. Any commutative algebra equipped with a derivation , that is, a commutative differential algebra, admits a Wronskian bracket under which it becomes a Lie algebra. Conversely, to any Lie algebra a commutative differential algebra is universally associated, its Wronskian envelope, in a way similar to the associative envelope. This contribution is the beginning of an investigation of these relations between Lie algebras and differential algebras which is parallel to the classical theory. In particular, we give a sufficient condition under which a Lie algebra may be embedded into its Wronskian envelope, and we present the construction of the free Lie algebra with this property.

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Section: Examplementioning

confidence: 99%

Wronskian Envelope of a Lie Algebra

Poinsot

2013

Algebra

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“…Closed form solutions to differential equations and the inclusion of special functions, possibly defined by lower order differential equations constitutes an active area of research. References are the book [van der Put and Singer 2003] and [Yuan and van Hoeij 2010], which has references to newer work. A connection to differential elimination theory of Section 2 should be noted.…”

Section: Computation Of Closed Form Solutionsmentioning

confidence: 99%

The “Seven Dwarfs” of Symbolic Computation

Kaltofen

2011

Texts &Amp; Monographs in Symbolic Computation

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We present the Seven Dwarfs of Symbolic Computation, which are sequential and parallel algorithmic methods that today carry a great majority of all exact and hybrid symbolic compute cycles. SymDwf 1. Exact linear algebra, integer lattices SymDwf 2. Exact polynomial and differential algebra, Gröbner bases SymDwf 3. Inverse symbolic problems, e.g., interpolation and parameterization SymDwf 4. Tarski's algebraic theory of real geometry SymDwf 5. Hybrid symbolic-numeric computation SymDwf 6. Computation of closed form solutions SymDwf 7. Rewrite rule systems and computational group theory We will elaborate on each dwarf and compare with Colella's seven and the Berkeley team's thirteen dwarfs of scientific computing.

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“…We first recall the structures of differential operators and polynomials; see for example [27,15] for more details. Let (F , ∂ ) be a commutative differential algebra over a field K, so ∂ : F → F is a K-linear map satisfying the Leibniz rule…”

Section: Differential Algebras Operators and Polynomialsmentioning

confidence: 99%