Small Height Points for Quadratic Polynomials

This page has links to all the data concerning Q-rational points of small height for quadratic polynomials, from the 2011 REU project with Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White. The project was funded mainly by NSF Grant DMS-0901494; in addition, Chen and Kovacheva were supported by Amherst College Dean of Faculty student funds.
The data concerns polynomials of the form z2 - c, with c lying between -10 and -3/4 and having denominator a square n2, where 1<n<60061, along with a point z with denominator n.

After discarding preperiodic points, the question is: how small can the canonical height of z (for the map z2 - c) be, relative to the (standard) height of c?

The following two comma-separated text files list all such pairs (c,z) where the ratio of the two heights is less than about 0.037:

   n not divisible by 4       n divisible by 4

Reading the data files:

Each line of each data file is in the format

c,z,[canonical height h_c(z) of z],[height ratio h_c(z)/h(c)]

Here is a sample entry, which appears in the ``n divisible by 4'' file:


This means the point 10541/21840 has canonical height about 0.30762 for the map z2 - 1013082841/476985600. Dividing this height by h(1013082841/476985600)= log(1013082841) gives the height ratio of about 0.014835.