 

John R. Doyle,
Paul Fili, and
Trevor Hyde
Dynatomic polynomials, necklace operators, and universal relations for dynamical units
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print


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


Abstract
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).


Acknowledgements
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 DMS2112697.
Trevor Hyde was partially supported by the NSF Postdoctoral Research Fellowship DMS2002176 and the Jump Trading Mathlab Research Fund.


Author information
John R. Doyle:
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA
john.r.doyle@okstate.edu
Paul Fili:
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078, USA
paul.fili@okstate.edu
Trevor Hyde:
Department of Mathematics
University of Chicago
Chicago, IL 60637, USA
tghyde@uchicago.edu

