I would like a proof or a reference (or a counter-example...) for the following fact. Let $P\in \mathbb{C}[x\_1,\ldots ,x\_n]$ and $D\in \mathbb{C}[\frac{\partial }{\partial x\_1} ,\ldots ,\frac{\partial }{\partial x\_n}]$ be nonzero homogeneous polynomials. Then there exists a homogeneous polynomial $Q\in \mathbb{C}[x\_1,\ldots ,x\_n]$ such that $D(Q)=P$. 



Surjectivity of differential operators with constant coefficients
Here is another approach. Let $R$ be a non-zero homogeneous polynomial of degree $n$. We want to show that the mapping $Q\mapsto R(\partial)Q$ is surgective from $V\_{m+n}$ to $V\_m$ where $V\_k$ is the space of homogeneous polynomials of degree $k$. Note that $\langle A,B\rangle=[A(\partial)\bar B](0)$ is a scalar product on $V\_k$ (with monomials forming an orthogonal basis). If our mapping is not surjective, then there exists a non-zero polynomial $S\in V\_m$ such that
$\langle S, R(\partial) Q\rangle=\langle S\bar R,Q\rangle=0$ for all $Q\in V\_{n+m}$. But $S\bar R$ is a non-zero polynomial, so taking $Q=S\bar R$, we get a contradiction.


