A Fischer type decomposition theorem from the apolar inner product

We continue the study initiated by H. S. Shapiro on Fischer decompositions of entire functions, showing that such decomposition exist in a weak sense (we do not prove uniqueness) under hypotheses regarding the order of the entire function $f$ to be expressed as $f=P\cdot q+r$, the polynomial $P$, and bounds on the apolar norm of homogeneous polynomials of degree $m$. These bounds, previously used by Khavinson and Shapiro, and by Ebenfelt and Shapiro, can be interpreted as a quantitative, asymptotic strengthening of Bombieri’s inequality. In the special case where both the dimension of the space and the degree of $P$ are two, we characterize for which polynomials $P$ such bounds hold.