  Consider a matrix $\mathbf{F} \in \mathbb{K}[x]^{m \times n}$ of univariate
polynomials over a field $\mathbb{K}$. We study the problem of computing the
column rank profile of $\mathbf{F}$. 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 $\mathbf{F}$ with a rank-sensitive
complexity of $O\tilde{~}(r^{\omega-2} n (m+D))$ operations in $\mathbb{K}$.
Here, $D$ is the sum of row degrees of $\mathbf{F}$, $\omega$ is the exponent
of matrix multiplication, and $O\tilde{~}(\cdot)$ hides logarithmic factors.
