矩阵相乘时特征多项式的确定性计算
Deterministic computation of the characteristic polynomial in the time of matrix multiplication
摘要：
本文描述了一种计算域上矩阵的特征多项式的算法，其渐近复杂性与两个方阵相乘的算法相同，直到常数因子。在此之前，这只能通过诉诸泛型假设或随机化技术来实现，而最著名的一般确定性算法的复杂性界限是由Keller-Gehrig在1985年获得的，并且涉及对数因子。我们的算法计算更一般的一元多项式矩阵的行列式在减少的形式，并依赖于新的子程序转换成转移的弱波波夫矩阵，转移的弱波波夫矩阵到转移的波波夫矩阵。
英文摘要：
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.
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