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 [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. 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.

Squared chaotic random variables: New moment inequalities with applications

“…As discussed below, inequality (1.3) contains important extensions and generalisations of estimates by Frenkel [12], Arias-de-Reyna [2] and Pinasco [21], connected, respectively, to the so-called real and complex polarization problem introduced in [9,24], and to lower bounds for products of homogenous polynomials. A discussion of these points is provided in the subsequent Sections 1.2 and 1.3.…”

“…where ·, · indicates the scalar product in C d . As a consequence, for d ≥ 2 and for every complex Hilbert space H of dimension at least d, one has that c d (H) = d d/2 , where the dth linear polarization constant is defined as A result of Pinasco [21,Theorem 5.3] further implies that one has equality in (1.7) if and only if the vectors x 1 , ..., x d are orthonormal; also, it is important to remark that the inequality (1.7) follows from K. Ball's solution of the complex plank problem -see [6]. The problem of explicitly computing linear polarization constants associated with real or complex Banach spaces dates back to the seminal papers [9,24].…”

Multiphoton states related via linear optics

“…The quantity we are investigating is known as the Bombieri norm of homogeneous polynomials [34] (in this case, f k ), which is known to be invariant under unitary rotations of the variables. This quantity can be expressed as an integral of |f ( a)| 2k over the (complex) unit sphere | a| = 1, [35,36] (equivalently, see [5,Lemma 15], where it is called Fock Inner Product).…”

Non-linear plank problems and polynomial inequalities

“…The authors also showed that this is the best universal constant, since there are polynomials on ℓ 1 for which we have equality. For complex Hilbert spaces and homogeneous polynomials, the second named author proved in [16] that the optimal constant is (3) M = (k 1 + · · · + k n ) k 1 +···+kn k k 1 1 · · · k kn n , when the dimension of the space is at least the number of polynomials. Using a complexification argument it is easy to find a constant for real Hilbert space from (3).…”