Totally real algebraic integers in short intervals, Jacobi polynomials, and unicritical families in arithmetic dynamics

TL;DR

This paper classifies all post-critically finite unicritical polynomials over the maximal totally real algebraic extension of Q, using new results involving Jacobi polynomials and bounds on algebraic integers with real embeddings.

Contribution

It introduces a classification of certain unicritical polynomials over a special number field and develops new tools involving Jacobi polynomials and algebraic integer bounds.

Findings

Complete classification of post-critically finite unicritical polynomials over the maximal totally real extension.

A recursion formula for the n-diameter of an interval based on Jacobi polynomial properties.

A numerical criterion providing bounds on algebraic integers with real embeddings.

Abstract

We classify all post-critically finite unicritical polynomials defined over the maximal totally real algebraic extension of ${\mathbb Q}$. Two auxiliary results used in the proof of this result may be of some independent interest. The first is a recursion formula for the $n$-diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion which allows one to the give a bound on the degree of any algebraic integer having all of its complex embeddings in a real interval of length less than $4$.