# Chebyshev coordinates and Salem numbers

## TL;DR

This paper explores how expressing polynomials in Chebyshev basis reveals families related to Salem numbers, providing methods to compute their root limits and identifying minimal polynomials of Salem numbers within these families.

## Contribution

It introduces a novel approach using Chebyshev polynomials to identify Salem number minimal polynomials and analyzes their root distributions.

## Key findings

- Certain hyperbolic polynomial families have roots in [-2,2]
- The span of these polynomial families exceeds 4
- Methods to compute the limits of extremal roots are provided

## Abstract

By expressing polynomials in the basis of Chebyshev polynomials, certain families of hyperbolic polynomials appear naturally. Some of these families have all their roots in the interval $[-2,2]$. In many cases the span of the family of polynomials thus found is greater than 4, and we show that they are the minimal polynomials of Salem numbers, possibly multiplied by some cyclotomic polynomials. In addition, we show how to compute the limit of the largest and smallest roots.