Lower bound for the Perron-Frobenius degrees of Perron numbers

TL;DR

This paper establishes a lower bound for the Perron-Frobenius degree of certain Perron numbers, demonstrating that some cubic Perron and biPerron numbers can have arbitrarily large degrees, advancing understanding of their algebraic properties.

Contribution

It provides a new lower bound for Perron-Frobenius degrees of Perron numbers that are not totally-real, and shows the existence of cubic and biPerron numbers with arbitrarily large degrees.

Findings

Existence of cubic Perron numbers with arbitrarily large Perron-Frobenius degrees

Existence of biPerron numbers with arbitrarily large Perron-Frobenius degrees

A new lower bound for Perron-Frobenius degrees of non-totally-real Perron numbers

Abstract

Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result known to Lind, McMullen and Thurston. A similar result is proved for biPerron numbers.