A Fischer type decomposition theorem from the apolar inner product

Abstract

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.