2021/01/29 by Markus Hunziker, Hunziker, Markus, Andrei Martı́nez-Finkelshtein +5
Computer Science · Mathematics · #14H50 #14N15 #30J10 #42C05 #47A12 #Algebraic Geometry (math.AG) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2101.12638
openalex publication_date 2021/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a natural number n≥3 and two points a and b in the unit disk \mathbb D in the complex plane, it is known that there exists a unique elliptical disk having a and b as foci that can also be realized as the intersection of a collection of convex cyclic n-gons whose vertices fill the whole unit circle \mathbb T. What is less clear is how to find a convenient formula or expression for such an elliptical disk. Our main results reveal how orthogonal polynomials on the unit circle provide a useful tool for finding such a formula for some values of n. The main idea is to realize the elliptical disk as the numerical range of a matrix and the problem reduces to finding the eigenvalues of that matrix.