shift ${\bm{s}} \in {\mathbb Z\xspace}^n$, $F$ is then an *${\bm{s}}$-minimal approximant basis* if it is ${\bm{s}}$-row reduced.

Let ${{\ensuremath{\mathsf{MinBasis}}\xspace}}(d,A,{\bm{s}})$ be a function that returns $(F,{\bm{\delta}})$, where $F$ is an ${\bm{s}}$-minimal approximant basis of $A$ of order $d$, and ${\bm{\delta}}  = {{\textnormal{rowdeg}}}_{{\bm{s}}} F$. The next lemma recalls a well known method of constructing minimal approximant bases recursively. Although the output of ${{\ensuremath{\mathsf{MinBasis}}\xspace}}$ may not be unique, the lemma holds for *any* ${\bm{s}}$-minimal approximant basis that ${{\ensuremath{\mathsf{MinBasis}}\xspace}}$ might return.

\[lem:paderec\] Let $A = \left [ \begin{array}{c|c}
A_1 & A_2 \end{array} \right ]$ over ${{\mathsf{K}}}[x]$. If $(F_1, {\bm{\delta}}_1) = {{\ensuremath{\mathsf{MinBasis}}\xspace}}(d,A_1,{\bm{s}})$ and $(F_2,{\bm{\delta}}_2) =
{{\ensuremath{\mathsf{MinBasis}}\xspace}}(d,F_1A_2,{\bm{\delta}}_1)$, then $F_2F_1$ is an ${\bm{s}}$-minimal approximant basis of $A$ of order $d$ with ${\bm{\delta}}_2 = {{\textnormal{rowdeg}}}_{{\bm{s}}} F_2 F_1$.

Sometimes only the [*negative part*]{} of an ${\bm{s}}$-minimal approximant bases is required, the submatrix of the approximant bases consisting of rows with negative ${\bm{s}}$-degree. Let function ${{\ensuremath{\mathsf{NegMinBasis}}\xspace}}(d,A,{\bm{s}})$ have the same output as ${{\ensuremath{\mathsf{MinBasis}}\xspace}}$, but with $F$ restricted to the negative part.

\[lem:paderecprune\] \[lem:paderec\] still holds if ${{\ensuremath{\mathsf{MinBasis}}\xspace}}$ is replaced by ${{\ensuremath{\mathsf{NegMinBasis}}\xspace}}$, and “an ${\bm{s}}$-minimal” is replaced with “the negative part of an ${\bm{s}}$-minimal.”

Using for example the algorithm `M-Basis` of [@giorgi_complexity_2003], it is easy to show that any order $d$ approximant 