Efficient decomposition of separable algebras

Authors:
W. Eberly;M. Giesbrecht
Affiliations:
Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4;Department of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Venue:
Journal of Symbolic Computation
Year:
2004

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Abstract

We present new, efficient algorithms for computations on separable matrix algebras over infinite fields. We provide a probabilistic method of the Monte Carlo type to find a generator for the center of a given algebra 2l ⊆ Fm × m over an infinite field F. The number of operations used is within a logarithmic factor of the cost of solving m × m systems of linear equations. A Las Vegas algorithm is also provided under the assumption that a basis and set of generators for the given algebra are available. These new techniques yield a partial factorization of the minimal polynomial of the generator that is computed, which may reduce the cost of computing simple components of the algebra in some cases.