[@DJK].

\[thm:tool2\] Let $P(x) = x^n + b_{n-1} x_{n-1} + \dots + b_0$ with $n\ge N$. Assume that

1.  $P(1/{\lambda}_i) = P'(1/{\lambda}_i) = \dots = P^{(k_i-1)}(1/{\lambda}_i) = 0$ for $i = 1, \dots, r$.

2.  $\sum_{j=0}^{n-1} |b_j| \leq 2$.

3.  There exists a non-singular $N \times N$ submatrix of the matrix $B$ (defined by (\[eq:B\]) below).

Then there exists a neighbourhood of $(\underbrace{0,0,\dots, 0}_{N})$ contained in $A$.

Let $$B_t(y) = \sum_{k=0}^t b_k y^{t-k}$$ for $t = 0, 1, \dots, n-1$.

Define the matrix $B$ as follows: $$\label{eq:B}
B := \begin{pmatrix}
B_{0}^{(k_1-1)}({\lambda}_1) & B_{1}^{(k_1-1)}({\lambda}_1) &  \dots & B_{{n-1}}^{(k_1-1)}({\lambda}_1) \\
     \vdots        &     \vdots         &         & \vdots \\
B_0^{(1)}({\lambda}_1) & B_1^{(1)}({\lambda}_1) &  \dots & B_{n-1}^{(1)}({\lambda}_1) \\
B_0({\lambda}_1)       & B_1({\lambda}_1)       &  \dots & B_{n-1}({\lambda}_1) \\
     \vdots        &     \vdots         &         & \vdots \\
     \vdots        &     \vdots         &         & \vdots \\
B_{0}^{(k_r-1)}({\lambda}_r) & B_{1}^{(k_r-1)}({\lambda}_r) &  \dots & B_{{n-1}}^{(k_r-1)}({\lambda}_r) \\
     \vdots        &     \vdots         &         & \vdots \\
B_0^{(1)}({\lambda}_r) & B_1^{(1)}({\lambda}_r) &  \dots & B_{n-1}^{(1)}({\lambda}_r) \\
B_0({\lambda}_r)       & B_1({\lambda}_r)       &  \dots & B_{n-1}({\lambda}_r) \\
\end{pmatrix}.$$ Here $B^{(s)}_t(y) = \frac{1}{s!} \frac{d^s}{d y^s} B_t(y)$. Notice that $B$ is an $N \times n$ matrix.

Let $P$ have the required properties and let $u_{-n}, \dots, u_{-1}$ satisfy $$\label{eq:un}
 \begin{pmatrix}x_1 \\ x_2 \\ 