Polynomials with many rational preperiodic points Article Swipe
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John R. Doyle
,
Trevor Hyde
·
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2201.11707
· OA: W4226307096
YOU?
·
· 2022
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2201.11707
· OA: W4226307096
In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in $\mathbb{Q}[x]$. We show that for all $d\geq 2$, there exists a polynomial $f_d(x) \in \mathbb{Q}[x]$ with $2\leq \mathrm{deg}(f_d) \leq d$ such that $f_d(x)$ has at least $d + \lfloor \log_2(d)\rfloor$ rational preperiodic points. Furthermore, we show that for infinitely many integers $d$, the polynomials $f_d(x)$ and $f_d(x) + 1$ have at least $d^2 + d\lfloor \log_2(d)\rfloor - 2d + 1$ common complex preperiodic points.
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