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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Lower bounds for norms of products of polynomials via Bombieri inequality
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by Damián Pinasco PDF
Trans. Amer. Math. Soc. 364 (2012), 3993-4010 Request permission

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
  • 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.
  • © Copyright 2012 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 364 (2012), 3993-4010
  • MSC (2010): Primary 30C10, 12D05, 26D05, 46G25
  • DOI: https://doi.org/10.1090/S0002-9947-2012-05403-1
  • MathSciNet review: 2912442