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 Sn(P ) = supQn [PQn]2, where P is a fixed m−homogeneous polynomial and Qn runs over the unit ball of the Hilbert space of n−homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on C 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 {zk}k=1 of unit vectors in a complex Hilbert space for which sup‖z‖=1 |〈z, z1〉 · · · 〈z, zn〉| is minimum must be an orthonormal system.