Computing points of small height for cubic polynomials

2008/07/03 by Robert L. Benedetto, Benjamin Dickman, Benedetto, Robert L. +9
Mathematics · #11G50 #11S99 #37F10 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0807.0468

openalex publication_date 2008/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height 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, 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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