## Involve: A Journal of Mathematics

• Involve
• Volume 2, Number 1 (2009), 37-64.

### Computing points of small height for cubic polynomials

#### Abstract

Let $ϕ∈ℚ[z]$ be a polynomial of degree $d$ at least two. The associated canonical height $ĥϕ$ is a certain real-valued function on $ℚ$ that returns zero precisely at preperiodic rational points of $ϕ$. 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, $ĥϕ$ is bounded below by a positive constant (depending only on $d$) times some kind of height of $ϕ$ 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.

#### Article information

Source
Involve, Volume 2, Number 1 (2009), 37-64.

Dates
Revised: 25 November 2008
Accepted: 26 November 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513799116

Digital Object Identifier
doi:10.2140/involve.2009.2.37

Mathematical Reviews number (MathSciNet)
MR2501344

Zentralblatt MATH identifier
1194.37187

#### Citation

Benedetto, Robert; Dickman, Benjamin; Joseph, Sasha; Krause, Benjamin; Rubin, Daniel; Zhou, Xinwen. Computing points of small height for cubic polynomials. Involve 2 (2009), no. 1, 37--64. doi:10.2140/involve.2009.2.37. https://projecteuclid.org/euclid.involve/1513799116

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