  Let $P$ be a fixed homogeneous polynomial. We present a sharp condition on
$P$ guaranteeing the existence of asymptotically larger bounds in Bombieri's
inequality, so for every homogeneous polynomial $q_m$ of degree $m$ we have
\begin{equation*} \left\Vert P q_{m}\right\Vert _{a}\geq C_{P} m^{l\left(
P\right) /2}\left\Vert q_{m}\right\Vert _{a}, \end{equation*} where $\| \cdot
\| _{a}$ denotes the apolar norm. Explicit estimates for $C_P > 0$ and $l(P) >
0$ are given.
