Publications [#10422] of Peter H Berman

Papers Published

1. Peter H. Berman, Computing the Galois Group of Y' = AY + B, Y' = AY Completely Reducible, Journal of Symbolic Computation [pdf]
   (last updated on 2002/12/19)

   Abstract:
   We consider a special case of the problem of computing the Galois group of a linear ordinary differential operator L in C(x)[D]. We assume that C is a computable, characteristic-zero, algebraically closed constant field with factorization algorithm. There exists a decision procedure, due to Compoint and Singer, to compute the group in case L is completely reducible. In ``Calculating the Galois group of L_1(L_2(y)) = 0, L_1, L_2 completely reducible operators,'' Berman and Singer address the case of a product of two completely reducible operators. Their article shows how to reduce that case to the case of an inhomogeneous equation of the form L(y) = b, L completely reducible, b in C(x). Their article further presents a decision procedure to reduce that inhomogeneous case to the case of the associated homogeneous equation L(y) = 0, using an algorithm whose steps include the computation of a certain set of factorizations of L; this set is very large and difficult to compute in general. In this article, we give a new algorithm to compute the Galois group of a system of first-order equations Y' = AY + B, A in Mat_n(C(x)), b in C(x)^n, in case the associated homogeneous system Y' = AY is completely reducible. After the additional step of applying a cyclic vector algorithm to translate between operators and first-order systems, our algorithm yields a more efficient method to compute the group of L(y) = b, L completely reducible, b in C(x). 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.