Let $P^k$ be the space of homogeneous polynomials of degree $k$, i.e. $P^k = \text{span} \{x_1^{k_1} \dots x_n^{k_n} : k_1 + k_2 + \dots + k_n = k\}$. I am trying to show that the Laplacian operator $\Delta : \mathcal{P}^k \to \mathcal{P}^{k-2}$ is surjective. I think I can use the identity $\Delta (f g) = (\Delta f)g + f (\Delta g) -2<\nabla f, \nabla g>$, but it seems doubtful.
Partial answer : Assume $p=\sum_{a+b=k} f_{a,b} x^a y^b$ such that $\Delta p = x^c y^d$ ($c+d=k-2$). $$\Delta p = \sum_{a+b=k} f_{a,b} \Delta(x^a y^b) = \sum_{a+b=k} f_{a,b} ((\Delta x^a) y^b+ x^a (\Delta y^b) -2 <\nabla x^a, \nabla y^b>)$$ $$=\sum_{a+b=k} f_{a,b} (a (a-1) x^{a-2} y^b+ b(b-1) x^a y^{b-2} = x^c y^d$$ ($<\nabla x^a, \nabla y^b>$ vanishes) and using induction on $n$.
Is it possible to find the good $p$ for surjectivity in that way? Otherwise, is there exists a nice way to prove surjectivity of $\Delta$? Could anyone be able to complete in details a proof for that exercise