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 $d\ge 2$, $M_d(X)\sim {\gamma }_d{X}^{-1}$ for an explicit constant ${\gamma }_d>0$. For $d\ge 4$, the error term is $O_{d,\epsilon }(X^{-2+\epsilon })$ for every $\epsilon >0$; for $d=3$ the error term is $O(X^{-3/2})$; and for $d=2$ the error term is $O(X^{-3/2}\log X)$.

arXiv comment13 pages, main theorem improved