Computing points of small height for cubic polynomials
Number Theory
2008-12-03 v3 Dynamical Systems
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.
Cite
@article{arxiv.0807.0468,
title = {Computing points of small height for cubic polynomials},
author = {Robert L. Benedetto and Benjamin Dickman and Sasha Joseph and Benjamin Krause and Daniel Rubin and Xinwen Zhou},
journal= {arXiv preprint arXiv:0807.0468},
year = {2008}
}
Comments
Final version, to appear in Involve. 20 pages