also to replace the $x^d$ moduli with arbitrary $g_1,\ldots,g_n \in {{\mathsf{K}}}[x]$. Formally, for any field ${{\mathsf{K}}}$:

\[prob:sim\_pade\] Given a tuple $({\bm{S}}, {\bm{g}}, {\bm{N}})$ where

-   ${\bm{S}} = (S_1,\ldots,S_n) \in {{\mathsf{K}}}[x]^n$ is a sequence of polynomials,

-   ${\bm{g}} = (g_1,\ldots,g_n) \in {{\mathsf{K}}}[x]^n$ is a sequence of moduli polynomials with $\deg S_i < \deg g_i$ for $i=1,\ldots,n$,

-   and ${\bm{N}} = (N_0,\ldots,N_n)
     \in {\mathbb Z\xspace}_{\geq 0}^{n+1}$ are degree bounds satisfying $1\leq N_0 \leq \max_i \deg g_i$ and $N_i \leq \deg g_i$ for $i=1,\ldots,n$,

find, if it exists, a non-zero vector $(\lambda, \phi_1, \ldots, \phi_n)$ such that

1.  $\lambda S_i \equiv \phi_i \mod g_i$ for $i = 1,\ldots, n$, and \[p1item1\]

2.  $\deg \lambda < N_0$ and $\deg \phi_i < N_i$ for $i=1,\ldots,n$.

We will call any vector $(\lambda, \phi_1, \ldots, \phi_n)$ as above *a solution* to a given Simultaneous [Padé]{}approximation problem. Note that if the $N_i$ are set too low, then it might be the case that no solution exists.

\[ex:simpade\] Consider over ${\mathbb F_{2}\xspace}[x]$ that $g_1 = g_2 = g_3 = x^5$, and ${\bm{S}} = (S_1,S_2,S_3) = 
      \left(x^{4} + x^{2} + 1,\,x^{4} + 1,\,x^{4} + x^{3} + 1\right)$, with degree bounds ${\bm{N}} = (5, 3, 4, 5)$. Then $\lambda_1 = x^4 