$\deg
\det A x^{{\bm{s}}} = \eta$. It follows that row $i$ of ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}$ must have degree at least $\eta - \eta_i$ since $\eta_i$ is the degree of column $i$ of $(A x^{{\bm{s}}}){^\top}$. However, entries in row $i$ of ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}$ are minors of the matrix obtained from $A x^{{\bm{s}}}$ by removing row $i$, hence have degree at most $\eta
- \eta_i$. It follows that the (row-wise) leading coefficient matrix of ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}$ is non-singular, hence ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}$ is row reduced. Since ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}=  
(\det x^{{\bm{s}}}) {{\textnormal{adj}}}(A){^\top}x^{-{\bm{s}}}$ we conclude that ${{\textnormal{adj}}}(A){^\top}$ is $(-{\bm{s}})$-row reduced with ${{\textnormal{rowdeg}}}_{(-{\bm{s}})} {{\textnormal{adj}}}(A) =
(\eta - \eta_1 - s, \ldots, \eta - \eta_n - s)$.

Minimal approximant bases
-------------------------

We recall the standard notion of minimal approximant basis, sometimes known as order basis or $\sigma$-basis [@beckermann_uniform_1994]. For a matrix $A \in {{\mathsf{K}}}[x]^{n \times m}$ and order $d \in
{\mathbb Z\xspace}_{\geq 0}$, an *order $d$ approximant* is a vector ${\bm{p}} \in
{{\mathsf{K}}}[x]^{1 \times n}$ such that ${\bm{p}}A \equiv {\bm{0}} \mod x^d.$ An *approximant basis of order $d$* is then a matrix $F \in
{{\mathsf{K}}}[x]^{n \times n}$ which is a basis of all order $d$ approximants. Such a basis always exists and has full rank $n$. For a 