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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lower bounds for norms of products of polynomials via Bombieri inequality

Author: Damián Pinasco
Journal: Trans. Amer. Math. Soc. 364 (2012), 3993-4010
MSC (2010): Primary 30C10, 12D05, 26D05, 46G25
Published electronically: March 21, 2012
MathSciNet review: 2912442
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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.

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Additional Information

Damián Pinasco
Affiliation: Departamento de Matemáticas y Estadística, Universidad Torcuato Di Tella, Miñones 2177 (C1428ATG), Ciudad Autónoma de Buenos Aires, Argentina – and – CONICET

Keywords: Bombieri’s inequality, Bombieri’s norm, Gaussian measure, plank problem, polynomial, product of linear functionals, uniform norm inequalities.
Received by editor(s): May 5, 2010
Received by editor(s) in revised form: June 15, 2010
Published electronically: March 21, 2012
Additional Notes: This work was partially supported by ANPCyT PICT 05 17-33042.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.