Chebyshev polynomials, Zolotarev polynomials and plane trees

TL;DR

This paper explores the relationships between Chebyshev and Zolotarev polynomials, introduces the concept of Z-homotopy for plane trees, and investigates geometric conditions and specific cases for trees with 5, 6, and 7 edges.

Contribution

It establishes necessary geometric conditions for Z-homotopy of plane trees and analyzes specific cases for trees with 5, 6, and 7 edges.

Findings

Z-homotopy defined for plane trees based on polynomial critical values

Characterization of Z-homotopy for trees with 5 and 6 edges

Analysis of an example in the class of trees with 7 edges

Abstract

A polynomial with exactly two critical values is called a generalized Chebyshev polynomial. A polynomial with exactly three critical values is called a Zolotarev polynomial. Two Chebyshev polynomials $f$ and $g$ are called Z-homotopic, if there exists a family $p_\alpha$, $\alpha\in [0,1]$, where $p_0=f$, $p_1=g$ and $p_\alpha$ is a Zolotarev polynomial, if $\alpha\in (0,1)$. As each Chebyshev polynomial defines a plane tree (and vice versa), Z-homotopy can be defined for plane trees. In this work we prove some necessary geometric conditions for plane trees Z-homotopy, describe Z-homotopy for trees with 5 and 6 edges and study one interesting example in the class of trees with 7 edges.