Polynomials with many rational preperiodic points

Doyle, John; Hyde, Trevor

doi:10.1090/tran/9406

Polynomials with many rational preperiodic points
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by John R. Doyle and Trevor Hyde;

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9406

Published electronically: May 8, 2025

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Abstract:

In this paper we study questions related to exceptional behavior of preperiodic points of polynomials and rational functions. We use techniques from the geometry of numbers and algebraic combinatorics to show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb {Q}[x]$ with $2\leq \deg (f_d) \leq d$ such that $f_d(x)$ has at least $d + \lfloor \log _2(d)\rfloor$ rational preperiodic points. Furthermore, we show that for all $d\geq 2$ there are polynomials $f(x)$ and $g(x)$ with at least $2d^2$ common complex preperiodic points.

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Polynomials with many rational preperiodic points
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