Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let
$$m = \min\{k \in \mathbb{N} \mid \exists P \in \mathscr{V}(A) \text{ with } \deg(P) = k\}$$
Show that there exists a unique polynomial $p \in \mathbb{C}[X]$ satisfying the three conditions\\
(i) $p \in \mathscr{V}(A)$,\\
(ii) $\deg(p) = m$,\\
(iii) $p$ is monic.