  Let $P_{2k}$ be a homogeneous polynomial of degree $2k$ and assume that there
exist $C>0$, $D>0$ and $\alpha \ge 0$ such that \begin{equation*} \left\langle
P_{2k}f_{m},f_{m}\right\rangle_{L^2(\mathbb{S}^{d-1})}\geq \frac{1}{C\left(
m+D\right) ^{\alpha }}\left\langle f_{m},f_{m}\right\rangle_{\mathbb{S}^{d-1}}
\end{equation*} for all homogeneous polynomials $f_{m}$ of degree $m.$ Assume
that $P_{j}$ for $j=0, \dots ,\beta <2k$ are homogeneous polynomials of degree
$j$. The main result of the paper states that for any entire function $f$ of
order $% \rho <\left( 2k-\beta \right) /\alpha $ there exist entire functions
$q$ and $h$ of order bounded by $\rho$ such that \begin{equation*} f=\left(
P_{2k}-P_{\beta }- \dots -P_{0}\right) q+h\text{ and }\Delta ^{h}r=0.
\end{equation*} This result is used to establish the existence of entire
harmonic solutions of the Dirichlet problem for parabola-shaped domains on the
plane, with data given by entire functions of order smaller than $\frac{1}{2}$.
