Involve: A Journal of Mathematics

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

Computing points of small height for cubic polynomials

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

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Let ϕ[z] be a polynomial of degree d at least two. The associated canonical height ĥϕ is a certain real-valued function on 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G50: Heights [See also 14G40, 37P30]
Secondary: 11S99: None of the above, but in this section 37F10: Polynomials; rational maps; entire and meromorphic functions [See also 32A10, 32A20, 32H02, 32H04]

canonical height $p$-adic dynamics preperiodic points


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.

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