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

Authors:

Institution:

(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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