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In this paper we are concerned with the Bohnenblust--Hille type inequalitiesfor certain polynomials of bounded degree but of very large number ofvariables. As the polynomials will be defined on groups, one can think aboutthe problem as the estimate of Sidon constants. In most cases the sharpconstants are unknown. We estimate the universal constant concerning the Sidontype estimates of degree $d$ polynomials of $n$ variables $z_1, \dots, z_n$with unimodular coefficients. For polynomials that have constant absolute valueof coefficients, this allows us to improve the estimate from \cite{DGMS}. Themain result is Theorem 1.7 below.

An estimate of Sidon constant for complex polynomials with unimodular coefficients