# Extremal problems for polynomials with real roots

## TL;DR

This paper investigates extremal polynomials with real roots under fixed discriminant constraints, identifying explicit families that minimize or maximize polynomial values at points off the real line, with applications to lemniscates and charge distributions.

## Contribution

The paper introduces two explicit families of extremal polynomials related to fixed discriminant and explores their properties, including explicit root and value formulas, extending classical polynomial extremal problems.

## Key findings

- Identified extremal polynomials as linear combinations of powers of linear functions.
- Connected extremal polynomials to generalized Jacobi (Gegenbauer) polynomials.
- Applied results to lemniscates and discrete charge energy problems.

## 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.