Towards an efficient Meat-axe algorithm using f-cyclic matrices: The density of uncyclic matrices in [formula omitted

Abstract

An element X in the algebra M ( n , F ) of all n × n matrices over a field F is said to be f-cyclic if the underlying vector space considered as an F [ X ] -module has at least one cyclic primary component. These are the matrices considered to be “good” in the Holt–Rees version of Norton's irreducibility test in the Meat-axe algorithm. We prove that, for any finite field F q , the proportion of matrices in M ( n , F q ) that are “not good” decays exponentially to zero as the dimension n approaches infinity. Turning this around, we prove that the density of “good” matrices in M ( n , F q ) for the Meat-axe depends on the degree, showing that it is at least 1 − 2 q ( 1 q + 1 q 2 + 2 q 3 ) n for q ⩾ 4 . We conjecture that the density is at least 1 − 1 q ( 1 q + 1 2 q 2 ) n for all q and n, and confirm this conjecture for dimensions n ⩽ 37 . Finally we give a one-sided Monte Carlo algorithm called Is f Cyclic to test whether a matrix is “good,” at a cost of O ( Mat ( n ) log n ) field operations, where Mat ( n ) is an upper bound for the number of field operations required to multiply two matrices in M ( n , F q ) .