Vol. 2, No. 1, 2009

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ISSN: 1944-4184 (e-only)
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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 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.

Keywords
canonical height, $p$-adic dynamics, preperiodic points
Mathematical Subject Classification 2000
Primary: 11G50
Secondary: 11S99, 37F10
Milestones
Received: 25 September 2008
Revised: 25 November 2008
Accepted: 26 November 2008
Published: 18 March 2009

Communicated by Bjorn Poonen
Authors
Robert L. Benedetto
Amherst College
Department of Math and Computer Science
31 Quadrangle Drive
Amherst, MA 01002
United States
http://www.cs.amherst.edu/~rlb
Benjamin Dickman
Amherst College
Department of Math and Computer Science
31 Quadrangle Drive
Amherst, MA 01002
United States
Sasha Joseph
Amherst College
Department of Math and Computer Science
31 Quadrangle Drive
Amherst, MA 01002
United States
Benjamin Krause
Amherst College
Department of Math and Computer Science
31 Quadrangle Drive
Amherst, MA 01002
United States
Daniel Rubin
Johns Hopkins University
Department of Mathematics
3400 N. Charles St
Baltimore, MD 21218
United States
Xinwen Zhou
Amherst College
Department of Math and Computer Science
31 Quadrangle Drive
Amherst, MA 01002
United States