Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2004; 41(1): 209-229

Printed January 1, 2004

Copyright © The Korean Mathematical Society.

Maximum moduli of unimodular polynomials

Andreas Defant, Domingo Garcia, and Manuel Maestre

Fachbereich Mathematik, Universitaet, Universidad de Valencia, Universidad de Valencia

Abstract

Let $\sum_{|\alpha|=m} s_{\alpha}z^\alpha$, $z \in \mathbb C^n$ be a unimodular $m$-homoge-neous polynomial in $n$ variables (i.e., $|s_\alpha|=1$ for all multi indices $\alpha$), and let $R \subset \mathbb C^n$ be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modulus $\sup_{z\in R}| \sum_{|\alpha|=m}{\hskip-0.05cm} s_{\alpha}z^\alpha|$, and upper estimates for the average of these maximum moduli taken over all possible $m$-homogeneous Bernoulli polynomials (i.e., $s_\alpha=\pm 1$ for all multi indices $\alpha$). Examples show that for a fixed degree $m$ our estimates, for rather large classes of domains $R$, are asymptotically optimal in the dimension $n$.

Keywords: several complex variables, power series, polynomials, Banach spaces, unconditional basis, Banach-Mazur distance

MSC numbers: Primary 32A05; Secondary 46B07, 46B09, 46G20