Given a Banach space $E$ and positive integers $k$ and $l$ we investigate the smallest constant $C$ that satisfies $\|P\|\hskip1pt\|Q\|\le C\|PQ\|$ for all $k$-homogeneous polynomials $P$ and $l$-homogeneous polynomials $Q$ on $E$. Our estimates are obtained using multilinear maps, the principle of local reflexivity and ideas from the geometry of Banach spaces (type and uniform convexity). We also examine the analogous problem for general polynomials on Banach spaces.