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

We consider a special case of the problem of computing the Galois group of a system of linear ordinary differential equations Y′ = MY, 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) address the case in which M = yjsco5390x.gif M 1 * 0 M 2 ], Y′ = M_iY completely reducible for i = 1, 2. Their article shows how to reduce that case to the case of an inhomogeneous system Y′ = AY + B, A C (x)^{n × n}, B C (x)ⁿ, Y′ = AY completely reducible. 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 L(y) = b,L C(x) D ], b C (x), then computing a certain set of factorizations of L in C(x)D ]; 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 systemY′ = 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.