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