matrix $P$ such that $$\hat{F} := PFP{^\top}= \left [ \begin{array}{c|c}
                             F_1 & \\\hline
                             F_2 & I_{n-k}
                           \end{array} \right ] \ ,$$ with each column of $F_1$ having degree strictly greater than zero. Let $i$ be the row index of the single 1 in the first column of $P{^\top}$. Since $F^{-1} = P{^\top}\hat{F}^{-1}P$, we have $$\label{first}
{\rm row}({{\textnormal{adj}}}(F),1)P^{-1} = x^D\, {\rm row}(\hat{F}^{-1},i).$$ Considering that $$\hat{F}^{-1} = \left [ \begin{array}{c|c} F_1^{-1} & \\\hline -F_2F_1^{-1} & I_{n-k} \end{array} \right ],$$ it will suffice to compute the first $k$ entries of the vector on the right hand side of (\[first\]). If $i \leq k$ then let ${\bm{v}} \in {{\mathsf{K}}}[x]^{1 \times k}$ be row $i$ of $x^{D}I_k$. Otherwise, if $i>k$ then let ${\bm{v}}$ be row $i-k$ of $-x^{D}F_2$. Then in both cases, ${\bm{v}}F_1^{-1}$ will be equal to the first $k$ entries of the vector on the right hand side of (\[first\]). Like before, high-order lifting combined with partial linearisation will compute this vector in $O\big(k^{\omega}\, {{\mathsf{M}}}(\lceil D/k \rceil)\,(\log \lceil D/k \rceil)
\big)$ operations in ${{\mathsf{K}}}$. Since $D\geq k$ the cost estimate remains valid if $\lceil D/k \rceil$ is replaced with $D/k$.

Reduction to Hermite [PADÉ]{} {#sec:dual}
=============================

In this section we present an algorithm for 