Extremal problems for polynomials with real roots

Abstract

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

Weighted Fekete Points on the Real Line and the Unit Circle

“…The weighted (or elliptic) equilibrium measure is known in this case as the arctan distribution, see Theorem 3 in [6]. The fact of weak* convergence of the counting measures for the weighted Fekete points to the arctan distribution was directly observed in Corollary 10 of [4]. These facts can be summarized as follows.…”

“…Proofs for Section 1. We start with an estimate for a certain product of sine functions contained in [4]. For convenience, we will give a short proof.…”

“…The following lemma is extracted from the proofs of Theorems 12 and 13 in [4]. We give a short proof for completeness.…”