of an ${\bm{h}}$-minimal approximant basis of $$H = \left [ \begin{array}{c} -{\bm{S}} \\
 I \\
 {  \ifx\null\null  \Gamma_{{\bm{g}}}  \else  \Gamma_{{\bm{g}}_{\null}}  \fi}\end{array} \right ] \in {{\mathsf{K}}}[x]^{(2n+1) \times n}$$ of order $d$, then the submatrix of $G$ comprised of the first $n+1$ columns is a solution basis to the problem instance.

An approximant ${\bm{v}} = (\lambda, \phi_1,\ldots,\phi_n,q_1,\ldots, q_n)$ of order $d$ of $H$ clearly satisfies $$\begin{aligned}
 \label{eq:lambda} 
\lambda S_i = \phi_i + q_ig_i \bmod
x^d\end{aligned}$$ for $i=1,\ldots,n$; conversely, any such vector ${\bm{v}}$ satisfying these congruences must be an approximant of $H$ of order $d$.

Now suppose ${\bm{v}}$ is an order $d$ approximant of $H$ with negative ${\bm{h}}$-degree, so $\deg \lambda \leq N_0-1$, $\deg \phi_i \leq
N_i-1$, and $\deg q_i \leq  N_0 - 2$. Since \[prob:sim\_pade\] specifies that $\deg S_i < \deg g_i$ and $N_i \leq \deg g_i$, both $\lambda S_i$ and $q_i g_i$ will have degree bounded by $N_0 + \deg g_i - 2$. Since \[prob:sim\_pade\] specifies that $N_0 \geq 1$, it follows that both the left and right hand sides of (\[eq:lambda\]) have degree bounded by $N_0+ \deg g_i -2$, which is strictly less than $d$. We conclude that $$\begin{aligned}
 \label{eq:lambda2}
\lambda S_i = \phi_i + q_i g_i\end{aligned}$$ for $i=1,\ldots,n$. It 