Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$ and we denote by $P_C$ the restriction of $P$ to the circle $C(0,1)$.
Show that the application
$$\phi_{m-2} : \left|\begin{array}{rll} \mathcal{P}_{m-2} & \rightarrow & \mathcal{P} \\ Q & \mapsto & \Delta \tilde{Q} \end{array}\right. \quad \text{where} \quad \tilde{Q}(x,y) = (1 - x^2 - y^2) Q(x,y)$$
is linear and injective and that $\operatorname{Im} \phi_{m-2} \subset \mathcal{P}_{m-2}$.