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

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 $d\ge 3$ and every real number $C>4\sqrt{2}$, $M_d(X){\ll }_d{X}^{-1}\exp (C\sqrt{\log X\mathrm{/}\log \log X})$. For $d=2$, we prove the optimal bound $M_2(X)\ll X^{-1}$.