The average number of rational preperiodic points of polynomials over $\mathbb{Q}$

Abstract

Let $M_d(X)$ denote the average number of rational preperiodic points among degree-$d$ polynomials over $\mathbb{Q}$ with vanishing $z^{d-1}$ coefficient, constant term $1$, and height at most $X$. We prove that, for every integer $d\ge2$, $M_d(X) \sim \gamma_d X^{-1}$ for an explicit constant $\gamma_d>0$. For $d\ge4$, the error term is $O_{d,\varepsilon}(X^{-2+\varepsilon})$ for every $\varepsilon>0$; for $d=3$ the error term is $O(X^{-3/2})$; and for $d=2$ the error term is $O(X^{-3/2}\log X)$.