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

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 $f(x)$.

19 pages, comments welcome!