On asymptotics of the uniform norm of polynomials with zeros at the roots of unity期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:

Affiliation:

(1) Department of Mathematics, St.-Petersburg State University Bibliotechnaya pl. 2, Peterhof, St.-Petersburg, 198504, Russia

Abstract:

The work is related to the problem by P. Erd odblac

s 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 radic

N, and we get estimates of possible values of the lower limit of the ratio (ln A_N/ radic

N as well as exact bounds of the upper limit of this ratio.

Keywords:

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