Lower bounds for norms of products of polynomials via Bombieri inequality

Abstract

In this paper we give a different interpretation of Bombieri’s norm. This new point of view allows us to work on a problem posed by Beauzamy about the behavior of the sequence $S_n(P)=\sup _{Q_n} [PQ_n]_2$, where $P$ is a fixed $m-$homogeneous polynomial and $Q_n$ runs over the unit ball of the Hilbert space of $n-$homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on $\mathbb {C}^N$ and we obtain sharp inequalities whenever the number of factors is no greater than $N$. In particular, we prove that for the product of homogeneous polynomials on infinite dimensional complex Hilbert spaces our inequality is sharp. Finally, we use these ideas to prove that any set $\{z_k\}_{k=1}^n$ of unit vectors in a complex Hilbert space for which $\sup _{\Vert z \Vert =1} \vert \langle z, z_1\rangle \cdots \langle z, z_n\rangle \vert$ is minimum must be an orthonormal system.