provided by the ${{\bm{s}}}$-Popov form. Although an ${{\bm{s}}}$-Popov form can be defined for a matrix of arbitrary shape and rank, it suffices here to consider the case of a non-singular matrix. The following definition is equivalent to [@jeannerod_computation_2016 Definition 1.2].

\[def:popov\] A non-singular matrix $R \in {{\mathsf{K}}}[x]^{n\times n}$ is in ${{\bm{s}}}$-Popov form if ${\rm LM}_{{\bm{s}}}(R)$ is unit lower triangular and the degrees of off-diagonal entries of $R$ are strictly less than the degree of the diagonal entry in the same column.

Adjoints of row reduced matrices
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For a non-singular matrix $A$ recall that the adjoint of $A$, denoted by ${\rm adj}(A)$, is equal to $(\det A)A^{-1}$, and that entry ${{{\textnormal{adj}}}(A)}_{i,j}{^\top}$ is equal to $(-1)^{i+j}$ times the determinant of the $(n-1) \times (n-1)$ sub-matrix that is obtained from $A$ by deleting row $i$ and column $j$.

\[lem:adjointRowReduced\] Let $A \in {{\mathsf{K}}}[x]^{n \times n}$ be ${\bm{s}}$-row reduced. Then ${{\textnormal{adj}}}(A){^\top}$ is $(-{\bm{s}})$-row reduced with $${{\textnormal{rowdeg}}}_{(-{\bm{s}})} {{\textnormal{adj}}}(A){^\top}=(\eta - s - \eta_1,\ldots , \eta - s -\eta_n) \ ,$$ where ${\bm{\eta}} = {{\textnormal{rowdeg}}}_{{\bm{s}}} A$, $\eta = \sum_i \eta_i$ and $s = \sum_i s_i$.

Since $A$ is ${\bm{s}}$-row reduced then $A x^{{\bm{s}}}$ is row reduced. Note that ${{\textnormal{adj}}}(A x^{{\bm{s}}}){^\top}(A x^{{\bm{s}}}){^\top}= (\det A x^{{\bm{s}}}) I_m$ with 