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arXiv · 2108.09333

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

John R. Doyle·Paul Fili·Trevor Hyde
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Abstract

Given 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 $f(x)$.

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John R. Doyle, Paul Fili, Trevor Hyde. 2021-08-20. Dynatomic polynomials, necklace operators, and universal relations for dynamical units. https://arxiv.org/abs/2108.09333

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