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Transactions of the American Mathematical Society
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
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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
Email: dpinasco@utdt.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05403-1
PII: S 0002-9947(2012)05403-1
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.