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

Consider a matrix ๐ โ๐[x]^m ร n of univariate polynomials over a field ๐. We study the problem of computing the column rank profile of ๐ . 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 ๐ with a rank-sensitive complexity of O(r^ฯ-2 n (m+D)) operations in ๐. Here, D is the sum of row degrees of ๐ , ฯ is the exponent of matrix multiplication, and O(ยท) hides logarithmic factors.

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