Monogenic Strictly-Perron Polynomials

TL;DR

This paper proves that for any degree n ≥ 2, there are infinitely many monogenic strictly-Perron polynomials, which are minimal polynomials of Perron numbers with specific algebraic properties.

Contribution

It establishes the existence of infinitely many monogenic strictly-Perron polynomials of any degree n ≥ 2, expanding understanding of algebraic integers and Perron numbers.

Findings

Existence of infinitely many such polynomials for each degree n≥2

Construction methods for these polynomials

Characterization of Perron numbers involved

Abstract

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. A strictly-Perron polynomial is the minimal polynomial of a Perron number $\lambda$ such that $\lambda$ is neither a Pisot number, an anti-Pisot number, nor a Salem number. For any natural number $n\ge 2$, we prove that there exist infinitely many monogenic strictly-Perron polynomials of degree $n$.