MAXIMUM MODULI OF UNIMODULAR POLYNOMIALS

Let $\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha},\;z\;{\in}\;{\mathbb{C}}^n$ be a unimodular m-homogeneous polynomial in n variables (i.e. $$\mid$s_{\alpha}$\mid$\;=\;1$ for all multi indices $\alpha$), and let $R\;{\subset}\;{\mathbb{C}}^n$ be a (bounded complete) Reinhardt domain. We give lower bounds for the maximum modules $sup_{z\;{\in}\;R\;$\mid$\Sigma_{$\mid$\alpha$\mid$=m}\;s_{\alpha}z^{\alpha}$\mid$$, 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.