Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix
George Labahn, Vincent Neiger, Thi Xuan Vu, Wei Zhou · 2022
Consider a matrix F ε K [x]^mxn of univariate polynomials over a field K. We study the problem of computing the column rank profile of F. To this end we first give an algorithm which improves the minimal kernel basis algorithm of Zhou, Labahn, and Storjohann (Proceedings ISSAC 2012). We then provide a second algorithm which computes the column rank profile of F with a rank-sensitive complexity of O~ (rw-2n(m+d)) operations in K. Here, D is the sum of row degrees of F, w is the exponent of matrix multiplication, and O~ (.) hides logarithmic factors.