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

TL;DR

This paper introduces a deterministic algorithm for computing the characteristic polynomial of a matrix over a field with asymptotic complexity comparable to matrix multiplication, eliminating the need for randomization or genericity assumptions.

Contribution

It presents a new deterministic algorithm that computes the characteristic polynomial in optimal asymptotic time, improving over previous methods that relied on randomization or genericity.

Findings

Algorithm matches matrix multiplication complexity

Computes determinant of polynomial matrices in reduced form

Introduces new subroutines for matrix transformations

Abstract

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, 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.