684 research outputs found
A Reduced Form for Linear Differential Systems and its Application to Integrability of Hamiltonian Systems
Let with be a differential linear
system. We say that a matrix is a {\em reduced
form} of if and there exists such that . Such a form is
often the sparsest possible attainable through gauge transformations without
introducing new transcendants. In this article, we discuss how to compute
reduced forms of some symplectic differential systems, arising as variational
equations of hamiltonian systems. We use this to give an effective form of the
Morales-Ramis theorem on (non)-integrability of Hamiltonian systems.Comment: 28 page
Zariski Closures of Reductive Linear Differential Algebraic Groups
Linear differential algebraic groups (LDAGs) appear as Galois groups of
systems of linear differential and difference equations with parameters. These
groups measure differential-algebraic dependencies among solutions of the
equations. LDAGs are now also used in factoring partial differential operators.
In this paper, we study Zariski closures of LDAGs. In particular, we give a
Tannakian characterization of algebraic groups that are Zariski closures of a
given LDAG. Moreover, we show that the Zariski closures that correspond to
representations of minimal dimension of a reductive LDAG are all isomorphic. In
addition, we give a Tannakian description of simple LDAGs. This substantially
extends the classical results of P. Cassidy and, we hope, will have an impact
on developing algorithms that compute differential Galois groups of the above
equations and factoring partial differential operators.Comment: 26 pages, more detailed proof of Proposition 4.
Hrushovski's Algorithm for Computing the Galois Group of a Linear Differential Equation
We present a detailed and simplified version of Hrushovski's algorithm that
determines the Galois group of a linear differential equation. There are three
major ingredients in this algorithm. The first is to look for a degree bound
for proto-Galois groups, which enables one to compute one of them. The second
is to determine the identity component of the Galois group that is the pullback
of a torus to the proto-Galois group. The third is to recover the Galois group
from its identity component and a finite Galois group.Comment: 27 page
A Characterization of Reduced Forms of Linear Differential Systems
A differential system , with
is said to be in reduced form if where
is the Lie algebra of the differential Galois group of
. In this article, we give a constructive criterion for a system to be in
reduced form. When is reductive and unimodular, the system is in
reduced form if and only if all of its invariants (rational solutions of
appropriate symmetric powers) have constant coefficients (instead of rational
functions). When is non-reductive, we give a similar characterization via
the semi-invariants of . In the reductive case, we propose a decision
procedure for putting the system into reduced form which, in turn, gives a
constructive proof of the classical Kolchin-Kovacic reduction theorem.Comment: To appear in : Journal of Pure and Applied Algebr
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