solving \[prob:sim\_pade\_basis\] when $g_1 = \ldots = g_n = x^d$ for some $d \in {\mathbb Z\xspace}_{\geq 0}$. The algorithm is based on the well-known duality between the Simultaneous [Padé]{}problem and the Hermite [Padé]{}problem, see for example [@beckermann_uniform_1992]. This duality, first observed in a special case [@Mahler68], and then later in the general case [@beckermann_recursiveness_1997], was exploited in [@beckermann_fraction-free_2009] to develop algorithms for the fraction free computation of Simultaneous [Padé]{}approximation. We begin with a technical lemma that is at the heart of this duality.

\[lem:duality\] Let $\hat A, \hat B \in {{\mathsf{K}}}[x]^{(n+1)\times(n+1)}$ be as follows. $$\begin{aligned}
    \hat A &= 
         \left [ \begin{array}{c|c}
                   x^d & -{\bm{S}} \\\hline
                       & I
                 \end{array} \right ]
    &&
    \hspace*{-1em}\hat B &= 
           \left [ \begin{array}{c|cccc}
                     1 & \\{  \hline  \\[\dimexpr-\normalbaselineskip+2pt]  }
                     {\bm{S}}{^\top}& x^d I
                   \end{array} \right ]
  \end{aligned}$$ Then $\hat B$ is the adjoint of $\hat A{^\top}$. Furthermore, $\hat A{^\top}$ is an approximant basis for $\hat
  B$ of order $d$, and $\hat B{^\top}$ is an approximant basis of $\hat A$ of order $d$.

Direct computation shows that $\hat A{^\top}\hat B = x^d I_m =
\det \hat A{^\top}I_m$, so $\hat B$ is the adjoint of $\hat
A{^\top}$.

Let now $G$ be an approximant basis of $\hat B$. By the 