Computing points of small height for cubic polynomials

Abstract

Let $\varphi \in \mathbb{Q}\left[z\right]$ be a polynomial of degree $d$ at least two. The associated canonical height ${ĥ}_{\varphi }$ is a certain real-valued function on $\mathbb{Q}$ that returns zero precisely at preperiodic rational points of $\varphi$. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on $d$. A related conjecture claims that at nonpreperiodic rational points, ${ĥ}_{\varphi }$ is bounded below by a positive constant (depending only on $d$) times some kind of height of $\varphi$ itself. In this paper, we provide support for these conjectures in the case $d=3$ by computing the set of small height points for several billion cubic polynomials.