a_{d1}^{i_d} \ \partial_{\beta=(i_1,\cdots,i_d)}^k f({\mathbf{x}}_0) \\
                    
                    \vdots       \\
                    
                    \partial_{d}^{k}f\circ A(A{^{\text{-}1}}{\mathbf{x}}_0) &=& \sum_{\sum i_j=k, i_j\geq 0} C_{(i_1,\cdots,i_d)} a_{1d}^{i_1}\cdots a_{dd}^{i_d} \ \partial_{\beta=(i_1,\cdots,i_d)}^k f({\mathbf{x}}_0). 
                    \end{array}$$ We want $A=(a_{ij})$ such that every element in the left side of (\[lin\_sys1\]) above is nonzero knowing that for at least one $\beta ,\  \partial_{ \beta=(i_1,\cdots,i_k)}^k f({\mathbf{x}}_0)\neq 0 $. Namely, we wish to find $A\in {\operatorname{GL}}_d(\mathcal{O}) $ such that $x'_i\neq 0$ for every $i$ where $$\begin{array}{rcr}
                    x'_1 &= & \sum C_{(i_1,\cdots,i_d)} \ a_{11}^{i_1}\cdots a_{d1}^{i_d} \ x_{(i_1,\cdots,i_d)}\\
                    \vdots   \\
                    x'_d &=& \sum C_{(i_1,\cdots,i_d)} \ a_{1d}^{i_1}\cdots a_{dd}^{i_d} \ x_{(i_1,\cdots,i_k)}
                        \end{array}$$ i.e. $$\begin{array}{rcr}
              x'_1&=& g(a_{11},\cdots,a_{d1})  \\
              \vdots \\
              x'_d&=& g(a_{1d},\cdots,a_{dd}) 
              \end{array}$$ and $g$ is a homogeneous polynomial of degree k. We already know that $ \partial_{ \beta=(i_1,\cdots,i_k)}^k f({\mathbf{x}}_0)\neq 0$ for at least one $\beta$, so at least one $x_{(i_1,\cdots,i_k)}\neq 0$ and thus $g $ is a nonzero polynomial.

Now $ g $ should have at least one nonzero value on $\{1+\pi\mathcal{O}\}\times\{\pi\mathcal{O}\} \times\cdots\times\{\pi\mathcal{O}\}$, otherwise $g$ is identically zero. So take $(a_{11},\cdots,a_{1d})$ to be the point of the aforementioned set where $g(a_{11},\cdots,a_{1d})\neq 0$. Then by a similar argument choose $(a_{i1},\cdots,a_{id})\in \{\pi\mathcal{O}\}\times\cdots\times\{1+\pi\mathcal{O}\} \times\cdots\times\{\pi\mathcal{O}\}$ such that $g(a_{i1},\cdots,a_{id})\neq 0$. Choosing $A$ this way we will automatically get that $\det(A)$ is a unit, which implies that $A\in {\operatorname{GL}}_d(\mathcal{O})$. 