Lower bounds for norms of products of polynomials via Bombieri inequality

Damián Pinasco
2012 Transactions of the American Mathematical Society  
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 [P Q 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 C N and we obtain sharp inequalities whenever the number of factors is no greater than N .
more » ... 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 } n k=1 of unit vectors in a complex Hilbert space for which sup z =1 | z, z 1 · · · z, z n | is minimum must be an orthonormal system.
doi:10.1090/s0002-9947-2012-05403-1 fatcat:o6wv5yjb2zgjbewnhmlhsfhi4i