Computing points of small height for cubic polynomials

Robert Benedetto, Benjamin Dickman, Sasha Joseph, Benjamin Krause, Daniel Rubin, Xinwen Zhou
2009 Involve. A Journal of Mathematics  
Let φ ∈ ‫[ޑ‬z] be a polynomial of degree d at least two. The associated canonical heightĥ φ is a certain real-valued function on ‫ޑ‬ 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,ĥ φ is bounded below by a positive constant (depending only on d) times some kind of height of φ itself. In
more » ... 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. MSC2000: primary 11G50; secondary 11S99, 37F10.
doi:10.2140/involve.2009.2.37 fatcat:zqukkhgewzegdnml6n2wgsbgfy