#### Vol. 2, No. 1, 2009

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Computing points of small height for cubic polynomials

### Robert L. Benedetto, Benjamin Dickman, Sasha Joseph, Benjamin Krause, Daniel Rubin and Xinwen Zhou

Vol. 2 (2009), No. 1, 37–64
##### Abstract

Let $\varphi \in ℚ\left[z\right]$ be a polynomial of degree $d$ at least two. The associated canonical height ${ĥ}_{\varphi }$ is a certain real-valued function on $ℚ$ 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.

##### Keywords
canonical height, $p$-adic dynamics, preperiodic points
##### Mathematical Subject Classification 2000
Primary: 11G50
Secondary: 11S99, 37F10