MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS

Abstract

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