Math @ Duke
Publications [#10422] of Peter H Berman
- Peter H. Berman, Computing the Galois Group of Y' = AY + B, Y' = AY Completely Reducible,
Journal of Symbolic Computation
(last updated on 2002/12/19)
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.
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