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arXiv · 0807.0468

Computing points of small height for cubic polynomials

Robert L. Benedetto·Benjamin Dickman·Sasha Joseph·Benjamin Krause·Daniel Rubin·Xinwen Zhou
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Abstract

Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height \hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. 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 non-preperiodic rational points, \hat{h}_f is bounded below by a positive constant (depending only on d) times some kind of height of f 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.

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Robert L. Benedetto, Benjamin Dickman, Sasha Joseph, Benjamin Krause, Daniel Rubin, Xinwen Zhou. 2008-12-03. Computing points of small height for cubic polynomials. https://arxiv.org/abs/0807.0468

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