We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and that there exists a basis $\mathcal{B} = (u_1, \ldots, u_n)$ of $E$ in which $f$ has a block diagonal matrix with blocks of size $m_k$ of the form
$$\left(\begin{array}{cccccc} \lambda_k & 0 & \cdots & \cdots & \cdots & 0 \\ 1 & \lambda_k & \ddots & & & \vdots \\ 0 & 1 & \lambda_k & \ddots & & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & \lambda_k & 0 \\ 0 & \cdots & \cdots & 0 & 1 & \lambda_k \end{array}\right)$$
We set $x_0 = u_1 + u_{m_1+1} + \cdots + u_{m_1 + \cdots + m_{p-1}+1}$.
Determine the polynomials $Q \in \mathbb{C}[X]$ such that $Q(f)(x_0) = 0$.