  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.
