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

Vincent Neiger (1) , Clément Pernet (2)

1 XLIM-MATHIS - Mathématiques & Sécurité de l'information
2 CASC - Calcul Algébrique et Symbolique, Sécurité, Systèmes Complexes, Codes et Cryptologie

This paper describes an algorithm which computes the characteristic polynomial of a matrix over a field within the same asymptotic complexity, up to constant factors, as the multiplication of two square matrices. Previously, to our knowledge, this was only achieved by resorting to genericity assumptions or randomization techniques, while the best known complexity bound with a general deterministic algorithm was obtained by Keller-Gehrig in 1985 and involves logarithmic factors. Our algorithm computes more generally the determinant of a univariate polynomial matrix in reduced form, and relies on new subroutines for transforming shifted reduced matrices into shifted weak Popov matrices, and shifted weak Popov matrices into shifted Popov matrices.

Mots clés: Characteristic polynomial; polynomial matrices; determinant; fast linear algebra

Domaines: Calcul formel [cs.SC]; Complexité [cs.CC]

Vincent Neiger, Clément Pernet. Deterministic computation of the characteristic polynomial in the time of matrix multiplication. Journal of Complexity, 2021, 67, pp.101572. ⟨10.1016/j.jco.2021.101572⟩. ⟨hal-02963147v2⟩