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Calculating the Galois Group of Y′ =   =   =  AY +   +   +  B,Y′ =   =   =  AY Completely Reducible

✍ Scribed by Peter Berman

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
291 KB
Volume
33
Category
Article
ISSN
0747-7171

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✦ Synopsis

We consider a special case of the problem of computing the Galois group of a system of linear ordinary differential equations Y = M Y , M ∈ C(x) n×n . We assume that C is a computable, characteristic-zero, algebraically closed constant field with a factorization algorithm. There exists a decision procedure, due to Compoint and Singer, to compute the group in case the system is completely reducible. Berman and Singer (1999, J. Pure Appl. Algebr., 139, 3-23)

completely reducible for i = 1, 2. Their article shows how to reduce that case to the case of an inhomogeneous system

Their article further presents a decision procedure to reduce this inhomogeneous case to the case of the associated homogeneous system Y = AY . The latter reduction involves using a cyclic-vector algorithm to find an equivalent inhomogeneous scalar equation

; this set is very large and difficult to compute in general.

In this article, we give a new and more efficient algorithm to reduce the case of a system Y = AY +B, Y = AY completely reducible, to that of the associated homogeneous system Y = AY . The new method's improved efficiency comes from replacing the large set of factorizations required by the Berman-Singer method with a single block-diagonal decomposition of the coefficient matrix satisfying certain properties.

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