TITLE: Surjectivity of differential operators with constant coefficients
QUESTION [8 upvotes]: 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 $D(Q)=P$.

REPLY [2 votes]: Too long for a comment. I want to use a version of the Lojaciewicz theorem of division of distributions by an analytic function (in fact Hörmander's result of division by a polynomial). We may assume that $P(x) =x^\alpha=x_1^{\alpha_1}\dots x_n^{\alpha_n}$  a monomial homogeneous with degree $\vert \alpha\vert=\alpha_1+\dots +\alpha_n$. Let us replace your notation $D$ by an operator $$A(D)=\sum_{\vert \beta\vert =m}a_\beta D^\beta,\quad D=-i\nabla.$$
The question at hand is to find an homogeneous polynomial $Q$ such that
$
A(D) Q= x^\alpha.
$
By Fourier transformation, it is equivalent to solve
$$
A(\xi)\widehat Q(\xi)=i^{\vert \alpha\vert}\delta^{(\alpha)},
$$
which means divide a derivative of the Dirac mass by an homogeneous ($A$ is assumed to be non-zero) polynomial. It is indeed possible by the aforementioned results of division and we find that $\widehat Q$ is homogeneous with degree $-n-\vert \alpha\vert-m$, so that $Q$ is homogeneous with degree $\vert \alpha\vert+m$. Moreover $\widehat Q$ is supported at the origin, proving that $Q$ is a polynomial.