An estimate of Sidon constant for complex polynomials with unimodular coefficients

TL;DR

This paper estimates the Sidon constant for complex polynomials with unimodular coefficients, improving bounds for polynomials of large degree and many variables, with implications for Bohnenblust--Hille inequalities.

Contribution

It provides a new universal estimate for the Sidon constant of degree d polynomials with unimodular coefficients, refining previous bounds.

Findings

Improved the estimate of the Sidon constant for certain polynomials.

Extended bounds to polynomials with large degrees and many variables.

Main result is Theorem 1.7 providing the new estimate.

Abstract

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