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Zbl pre05610504
Glasby, S.P.; Praeger, Cheryl E.
Towards an efficient meat-axe algorithm using $f$-cyclic matrices: The density of uncyclic matrices in M$(n,q)$. (English)
[J] J. Algebra 322, No. 3, 766-790 (2009). ISSN 0021-8693

This is a continuation of research of the meat-axe algorithm using $f$-cyclic matrices, see {\it S. P. Glasby} [J. Algebra 300, No.~1, 77--90 (2006; Zbl 1108.15015)]. The lower and upper bounds for the density of uncyclic matrices in $\text{M}(n,\Bbb{F}_q)$ are obtained. The authors give a practical Monte Carlo algorithm to test whether a given matrix is $f$-cyclic relative to some irreducible divisor of its characteristic polynomial. Also the algorithm outputs a witness vector which can be used when applying Norton's irreducibility test; see {\it D. F. Holt} and {\it S. Rees} [J. Aust. Math. Soc., Ser. A 57, No.~1, 1--16 (1994; Zbl 0833.20021)]. The work is written understandably for nonspecialists; the references to available programs are present.
[Serghey G. Suvorov (Donetsk)]

MSC 2000:: *65C05 Monte Carlo methods
15B05

Keywords: complexity analysis; meat-axe algorithm; $f$-cyclic matrices; uncyclic matrices; Monte Carlo algorithm; Norton's irreducibility test

Citations: Zbl 1108.15015; Zbl 0833.20021

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