Extremal problems for polynomials with real roots

JOURNAL OF APPROXIMATION THEORY

卷:253

Article

Dubickas, Arturas¹  Pritsker, Igor² 

[1] Vilnius Univ, Fac Math & Informat, Inst Math, Naugarduko 24, LT-03225 Vilnius, Lithuania

[2] Oklahoma State Univ, Dept Math, Stillwater, OK 74078 USA

关键词: Polynomials;    Real roots;    Discriminant;    Lemniscate;    Lagrange multipliers method;    Minimum energy problem;    Jacobi polynomial;   

DOI  :  10.1016/j.jat.2020.105376

来源: Elsevier

【 摘 要 】

We consider polynomials of degree d with only real roots and a fixed value of discriminant, and study the problem of minimizing the absolute value of such polynomials at a fixed point off the real line. There are two explicit families of polynomials that turn out to be extremal in terms of this problem. The first family has a particularly simple expression as a linear combination of dth powers of two linear functions. Moreover, if the value of the discriminant is not too small, then the roots of the extremal polynomial and the smallest absolute value in question can be found explicitly. The second family is related to generalized Jacobi (or Gegenbauer) polynomials, which helps us to find the associated discriminants. We also investigate the dual problem of maximizing the value of discriminant, while keeping the absolute value of polynomials at a point away from the real line fixed. Our results are then applied to problems on the largest disks contained in lemniscates, and to the minimum energy problems for discrete charges on the real line. (C) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

Free