Cyclotomic properties of polynomials associated with automatic sequences

TL;DR

This paper investigates the cyclotomic properties of polynomials linked to automatic sequences, revealing recurrence relations at roots of unity and analyzing their minimal order and coefficient integrality.

Contribution

It generalizes previous results on Rudin-Shapiro polynomials to a broader class of automatic sequence polynomials, exploring their recurrence relations and algebraic properties.

Findings

Polynomials satisfy specific recurrence relations at roots of unity.

The minimal order of these relations is characterized.

The integrality of the recurrence coefficients is studied.

Abstract

We show that polynomials associated with automatic sequences satisfy a certain recurrence relation when evaluated at a root of unity, which generalizes a result of Brillhart, Lomont and Morton on the Rudin--Shapiro polynomials. We study the minimal order of such a relation and the integrality of its coefficients.