Volume 56, Issue 1 p. 364-379
RESEARCH ARTICLE

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

Chatchai Noytaptim

Chatchai Noytaptim

Department of Mathematics, Oregon State University, Corvallis, Oregon, USA

Search for more papers by this author
Clayton Petsche

Corresponding Author

Clayton Petsche

Department of Mathematics, Oregon State University, Corvallis, Oregon, USA

Correspondence

Clayton Petsche, Clayton Petsche, Department of Mathematics, Oregon State University, Corvallis, OR 97331, USA.

Email: [email protected]

Search for more papers by this author
First published: 17 October 2023

Abstract

We classify all postcritically finite unicritical polynomials defined over the maximal totally real algebraic extension of Q ${\mathbb {Q}}$ . 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 th $n {\rm th}$ diameter of an interval, which uses properties of Jacobi polynomials. The second is a numerical criterion that allows one to 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.