New York Journal of Mathematics
Volume 28 (2022), 534-556


John R. Doyle, Paul Fili, and Trevor Hyde

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

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Published: March 9, 2022.
Keywords: dynamical units, preperiodic points, necklace polynomials.
Subject: 37P05, 11R27, 37P35.

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

Author information

John R. Doyle:
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA


Paul Fili:
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA


Trevor Hyde:
Department of Mathematics
University of Chicago
Chicago, IL 60637, USA