For a generic polynomial f(x), the generalized dynatomic polynomial Φ_f,c,d(x) vanishes at precisely those α such that f^c(α) has period exactly d under iteration of f(x). We show that the shifted dynatomic polynomials Φ_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 Φ_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).

We are happy to thank Valentin Huguin, Rafe Jones, Patrick Morton, and Joe Silverman for feedback and corrections on an earlier draft. We also thank the anonymous referee for helpful comments. John Doyle was partially supported by NSF grant DMS-2112697. Trevor Hyde was partially supported by the NSF Postdoctoral Research Fellowship DMS-2002176 and the Jump Trading Mathlab Research Fund.