%% Papers Published
@article{fds9870,
Author = {Peter H. Berman and M. F. Singer},
Title = {Calculating the Galois Group of L_1(L_2(y))=0, L_1, L_2
Completely Reducible Operators},
Journal = {Journal of Pure and Applied Algebra, vol. #139 (1999), pp.
324},
url = {http://www4.ncsu.edu:8030/~singer/papers/12Inhom.ps},
Abstract = {This article addresses the problem of computing the Galois
group of a product of two completely reducible linear
operators with rational function coefficients. It shows how
to reduce this problem to the case of a single completely
reducible linear operator, which is addressed in a separate
article by Compoint and Singer. A decision procedure and
several examples are provided.},
Key = {fds9870}
}
@article{fds10422,
Author = {Peter H. Berman},
Title = {Computing the Galois Group of Y' = AY + B, Y' = AY
Completely Reducible},
Journal = {Journal of Symbolic Computation},
url = {http://www.math.duke.edu/~berman/Y_inhom06.pdf},
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,
characteristiczero, 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 firstorder 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
firstorder 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 BermanSinger method with a single
blockdiagonal decomposition of the coefficient matrix
satisfying certain properties.},
Key = {fds10422}
}
