On asymptotics of the uniform norm of polynomials with zeros at the roots of unity

摘要

The work is related to the problem by P. Erdős about the estimation of the numbers $$A_N = _{left| z right| = 1}^{max } left| {left( {z - z_1 } right)left( {z - z_2 } right) cdots left( {z - z_N } right)} right|,{text{ where }}left| {z_j } right| equiv 1,$$ as N→∞.We shall deal with the case where the z j are all possible roots of the unity ordered in such a way that all the roots of degree n follow the roots of degree n-1. Within the group of roots of degree n, the enumeration can vary. We prove that ln A N grows like √N, and we get estimates of possible values of the lower limit of the ratio (ln A N /√N as well as exact bounds of the upper limit of this ratio.