Asympotic bounds for Bombieri's inequality on products of homogeneous polynomials

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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_a$ denotes the apolar norm. Explicit estimates for $C_P>0$ and $l(P)>0$ are given.