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This paper describes an algorithm which computes the characteristicpolynomial of a matrix over a field within the same asymptotic complexity, upto constant factors, as the multiplication of two square matrices. Previously,this was only achieved by resorting to genericity assumptions or randomizationtechniques, while the best known complexity bound with a general deterministicalgorithm was obtained by Keller-Gehrig in 1985 and involves logarithmicfactors. Our algorithm computes more generally the determinant of a univariatepolynomial matrix in reduced form, and relies on new subroutines fortransforming shifted reduced matrices into shifted weak Popov matrices, andshifted weak Popov matrices into shifted Popov matrices.

Deterministic computation of the characteristic polynomial in the time of matrix multiplication