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.

中文翻译：

---

矩阵乘法时特征多项式的确定性计算

本文描述了一种算法，该算法在相同渐近复杂度的域内计算矩阵的特征多项式，最多为常数因子，作为两个方阵的乘法。以前，据我们所知，这只能通过采用通用性假设或随机化技术来实现，而众所周知的与通用确定性算法绑定的复杂性是由 Keller-Gehrig 在 1985 年获得的，并且涉及对数因子。我们的算法更一般地计算简化形式的单变量多项式矩阵的行列式，并依赖于新的子程序将移位简化矩阵转换为移位弱波波夫矩阵，并将弱波波夫矩阵转换为移位波波夫矩阵。