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arXiv · 2409.18074

On the number of quadratic polynomials with a given portrait

Ho Chung Siu
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Abstract

Let $F$ be a number field. Given a quadratic polynomial $f_c(z) = z^2 + c \in F[z]$, we can construct a directed graph $Preper(f_c, F)$ (also called a portrait), whose vertices are $F$-rational preperiodic points for $f_c$, with an edge $\alpha \to \beta$ if and only if $f_c(\alpha) = \beta$. Poonen and Faber classified the portraits that occur for infinitely many $c$'s. Given a portrait $P$, we prove an asymptotic formula for counting the number of $c \in F$'s by height, such that $Preper(f_c, F) \cong P$. We also prove an asymptotic formula for the analogous counting problem, where $Preper(f_c, K) \cong P$ for some quadratic extension $K/F$. These results are conditioned on Morton-Silverman conjecture.

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Ho Chung Siu. 2024-09-26. On the number of quadratic polynomials with a given portrait. https://arxiv.org/abs/2409.18074

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