Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$. Deduce that there exists a polynomial $T \in \mathcal{P}_{m-2}$ such that $P + (1 - x^2 - y^2) T$ is a harmonic polynomial.
Let $m$ be an integer greater than or equal to 2. We consider a polynomial $P \in \mathcal{P}_m$.
Deduce that there exists a polynomial $T \in \mathcal{P}_{m-2}$ such that $P + (1 - x^2 - y^2) T$ is a harmonic polynomial.