A remark on polynomials and the transfinite diameter

Erdös, P.; Netanyahu, E.

doi:10.1007/bf02761531

Cited by 2 publications

(5 citation statements)

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“…It is immediate to see that ( 27) is equivalent to (9), and (28) is equivalent to (10). Hence the first part of this theorem follows from Theorem 4.…”

Section: This Is Equivalent To 2rmentioning

confidence: 73%

“…It follows from (1) that (9). Assume that (10) is true, and that equality holds in (9). Then 1) , which yields (2) for the polynomial f .…”

Section: Polynomials G Aλ In Terms Of Jacobi and Gegenbauer Polynomialsmentioning

confidence: 94%

“…then equality in (9) is attained if and only if f (x) = F (x) or f (x) = (−1) d F (−x), where F = F a,B is defined in (3) and its roots are given by (5).…”

Section: Extremal Problems and Their Solutionsmentioning

confidence: 99%

“…The authors also asked a number of questions related to the rate of this decay and the size of the largest disk contained in E(f ). Erdős and Netanyahu [9] proved that if all the roots of a monic polynomial f of degree d are contained in a fixed compact connected set of transfinite diameter (logarithmic capacity) c < 1, then E(f ) contains a disk of radius r c that depends only on c. The assumptions of Erdős and Netanyahu imply that lim sup d→∞ ∆…”

Section: Applicationsmentioning

confidence: 99%

“…Consider any monic polynomial f of degree d with d real roots and discriminant ∆ f . It follows from (1) that (9). Assume that (10) is true, and that equality holds in (9).…”

Section: Polynomials G Aλ In Terms Of Jacobi and Gegenbauer Polynomialsmentioning

confidence: 99%

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Extremal problems for polynomials with real roots

Dubickas

Pritsker

2020

Journal of Approximation Theory

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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 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 d-th 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.Throughout, let K(d, D) be the set of monic polynomials of degree d with d real roots and discriminant D. In all what follows, we will investigate the following natural problem.Problem 1. Let a > 0, D > 0 and d ≥ 2. Find all f ∈ K(d, D) that realize the minimum of |f (ai)|.We also state the dual problem:Problem 2. Let a > 0, m > a d and d ≥ 2. Find all monic polynomials f of degree d with d real roots and fixed value |f (ai)| = m that have the largest possible value of discriminant.The value |f (ai)| for any monic polynomial f (x) = d k=1 (x − x k ) of degree d with real roots is easily estimated as follows:Equality holds above if and only if x k = 0 for all k = 1, . . . , d. Hence, it is natural to use the restriction m > a d in the statement of Problem 2.Our first theorem solves Problem 1 if a is not too large in terms of d and D.5

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“…It is immediate to see that ( 27) is equivalent to (9), and (28) is equivalent to (10). Hence the first part of this theorem follows from Theorem 4.…”

Section: This Is Equivalent To 2rmentioning

confidence: 73%

“…It follows from (1) that (9). Assume that (10) is true, and that equality holds in (9). Then 1) , which yields (2) for the polynomial f .…”

Section: Polynomials G Aλ In Terms Of Jacobi and Gegenbauer Polynomialsmentioning

confidence: 94%

“…then equality in (9) is attained if and only if f (x) = F (x) or f (x) = (−1) d F (−x), where F = F a,B is defined in (3) and its roots are given by (5).…”

Section: Extremal Problems and Their Solutionsmentioning

confidence: 99%

Section: Applicationsmentioning

confidence: 99%

“…Consider any monic polynomial f of degree d with d real roots and discriminant ∆ f . It follows from (1) that (9). Assume that (10) is true, and that equality holds in (9).…”

Section: Polynomials G Aλ In Terms Of Jacobi and Gegenbauer Polynomialsmentioning

confidence: 99%

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