LOWER BOUNDS FOR NORMS OF PRODUCTS OF POLYNOMIALS VIA BOMBIERI INEQUALITY

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) = supQ(n) [PQ(n)](2), where P is a fixed m-homogeneous polynomial and Q(n). runs over the unit hall of the Hilbert space of n-homogeneous polynomials. We also study the factor problem for homogeneous polynomials defined on 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)(n)=1 of unit vectors in a complex Hilbert space for which sup(parallel to z parallel to=1)vertical bar < z, z(1)> . . . < z, z(n)>vertical bar is minimum must be an orthonormal system.