Rank-Sensitive Computation of the Rank Profile of a Polynomial Matrix

Abstract

Consider a matrix F ∈ K 𝑚×𝑛 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˜(𝑟 𝜔−2 𝑛(𝑚 + 𝐷)) operations in K. Here, 𝐷 is the sum of row degrees of F, 𝜔 is the exponent of matrix multiplication, and 𝑂˜(•) hides logarithmic factors.