287 research outputs found
Computing the Galois group of some parameterized linear differential equation of order two
We extend Kovacic's algorithm to compute the differential Galois group of
some second order parameterized linear differential equation. In the case where
no Liouvillian solutions could be found, we give a necessary and sufficient
condition for the integrability of the system. We give various examples of
computation.Comment: 14 pages, final version. To appear in Proceedings of the American
Mathematical Societ
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.
Introduction to the Galois Theory of Linear Differential Equations
This is an expanded version of the 10 lectures given as the 2006 London
Mathematical Society Invited Lecture Series at the Heriot-Watt University 31
July - 4 August 2006.Comment: 82 pages; some typos correcte
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