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.
Categories: math.AP