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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
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Abstract |
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Let be a polynomial of degree at least two. The associated canonical height is a certain real-valued function on that returns zero precisely at preperiodic rational points of . Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on . A related conjecture claims that at nonpreperiodic rational points, is bounded below by a positive constant (depending only on ) times some kind of height of itself. In this paper, we provide support for these conjectures in the case by computing the set of small height points for several billion cubic polynomials. |
Keywords
canonical height, $p$-adic dynamics, preperiodic points
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Mathematical Subject Classification 2000
Primary: 11G50
Secondary: 11S99, 37F10
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Milestones
Received: 25 September 2008
Revised: 25 November 2008
Accepted: 26 November 2008
Published: 18 March 2009
Communicated by Bjorn Poonen
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Authors |
| Robert L. Benedetto | |
| Amherst College Department of Math and Computer Science 31 Quadrangle Drive Amherst, MA 01002 United States |
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| http://www.cs.amherst.edu/~rlb | |
| Benjamin Dickman | |
| Amherst College Department of Math and Computer Science 31 Quadrangle Drive Amherst, MA 01002 United States |
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| Sasha Joseph | |
| Amherst College Department of Math and Computer Science 31 Quadrangle Drive Amherst, MA 01002 United States |
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| Benjamin Krause | |
| Amherst College Department of Math and Computer Science 31 Quadrangle Drive Amherst, MA 01002 United States |
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| Daniel Rubin | |
| Johns Hopkins University Department of Mathematics 3400 N. Charles St Baltimore, MD 21218 United States |
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| Xinwen Zhou | |
| Amherst College Department of Math and Computer Science 31 Quadrangle Drive Amherst, MA 01002 United States |
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