Dynatomic polynomials, necklace operators, and universal relations for dynamical units

TL;DR

This paper explores the algebraic structure of dynatomic polynomials related to polynomial iteration, revealing new relations among dynamical units and their connection to necklace operators and cyclotomic factors.

Contribution

It demonstrates that shifted dynatomic polynomials often contain factors linked to necklace polynomials, establishing a novel connection between dynamical units and cyclotomic factors.

Findings

Dynatomic polynomials have factors related to necklace operators.

New multiplicative relations among dynamical units are identified.

Connections between dynatomic factors and cyclotomic polynomials are established.

Abstract

Given a generic polynomial $f(x)$, the generalized dynatomic polynomial $\Phi_{f,c,d}(x)$ vanishes at precisely those $\alpha$ such that $f^c(\alpha)$ has period exactly $d$ under iteration of $f(x)$. We show that the shifted dynatomic polynomials $\Phi_{f,c,d}(x) - 1$ often have generalized dynatomic factors, and that these factors are in correspondence with certain cyclotomic factors of necklace polynomials. These dynatomic factors of $\Phi_{f,c,d}(x) - 1$ have an interpretation in terms of new multiplicative relations between dynamical units which are uniform in the polynomial $f(x)$.