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Publications of Peter H Berman    :chronological  alphabetical  combined listing:

%% Papers Published   
@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,
             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.},
   Key = {fds10422}
}

@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.
             3-24},
   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}
}

 

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