# Caustics of Poncelet polygons and classical extremal polynomials

## TL;DR

This paper explores the relationship between periodic billiard trajectories within ellipses and extremal polynomials, providing new characterizations, classical connections, and a catalog of small-period trajectories using advanced algebraic and geometric tools.

## Contribution

It introduces a novel approach linking billiard caustics to extremal polynomials and classical elliptic curve theory, offering new characterizations and classifications of periodic trajectories.

## Key findings

- Connection between billiard caustics and extremal polynomials
- Complete catalog of small-period billiard trajectories
- New proofs of monotonicity of rotation numbers

## Abstract

A comprehensive analysis of periodic trajectories of billiards within ellipses in the Euclidean plane is presented. The novelty of the approach is based on a relationship recently established by the authors between periodic billiard trajectories and extremal polynomials on the systems of $d$ intervals on the real line and ellipsoidal billiards in $d$-dimensional space. Even in the planar case, systematically studied in the present paper it leads to new results in characterizing $n$ periodic trajectories vs. so-called $n$ elliptic periodic trajectories, which are $n$-periodic in elliptical coordinates. The characterizations are done both in terms of the underlying elliptic curve and divisors on it and in terms of polynomial functional equations, like Pell's equation. This new approach also sheds light on some classical results. In particular we connect search for caustics which generate periodic trajectories with three classical classes of extremal polynomials on two intervals, introduced by Zolotarev and Akhiezer. The main classifying tool are winding numbers, for which we provide several interpretations, including one in terms of numbers of points of alternance of extremal polynomials. The latter implies important inequality between the winding numbers, which as a consequence, provides another proof of monotonicity of rotation numbers. A complete catalog of billiard trajectories with small periods is provided for $n=3, 4, 5, 6$ along with an effective search for caustics. As a byproduct, an intriguing connection between Cayle type conditions and discriminantly separable polynomials has been observed for all those small periods.