Small Height Points for Quadratic Rational Functions

This page has links to all the data concerning Q-rational points of small height for quadratic rational functions, from the 2011 REU project with Ruqian Chen, Trevor Hyde, Yordanka Kovacheva, and Colin White. Further computations were done in Spring 2013. 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 pairs (x,f) where x is a point in P1(Q) and f is a quadratic rational function. The goal is to find pairs for which x is preperiodic under the iteration of f with a forward orbit of length at least 6, or non-preperiodic but of particularly small canonical height with respect to f. The space of all pairs (x,f) up to coordinate change can be parametrized by triples (x3,x4,x5) of distinct rational numbers, none equal to 0 or 1, and away from a certain degeneracy locus. (Simply move x to infinity, f(x) to 1, and f2(x) to 0; then x3, x4, and x5 are the next three iterates.) The data here list all triples of height less than log(100) (i.e., numerator and denominator of x3, x4, and x5 are each between -100 and 100) that have long preperiodic orbits or small canonical height. Here, ``small'' means that the canonical height of x with respect to f is at most 0.002 times h(f), where h(f) is the height of f as a point on the appropriate moduli space.

The following text files list all such preperiodic pairs (x,f) our search found, grouped by the length of the cycle period. Period 2, Length 6 had far too many examples and is omitted.

                                    Period 2, Length 8    Period 3, Length 8   

   Period 1, Length 7    Period 2, Length 7    Period 3, Length 7    Period 4, Length 7    Period 5, Length 7    Period 6, Length 7    Period 7, Length 7   

   Period 1, Length 6    (Per 2, Len 6 omitted)   Period 3, Length 6    Period 4, Length 6    Period 5, Length 6    Period 6, Length 6   

The following text file lists all such pairs (x,f) where the ratio of the two heights is less than about 0.002. Within the file, pairs appear in order of increasing height ratio.


The following text file is the original raw data, i.e., the union of the above files, but in the order the search found them, i.e., unsorted.

   Raw Data   

Reading the data files:

Here is a sample entry from the Period 4 file:

phi(w) = (-1925*w^2 + 5071*w - 3146)/(-1925*w^2 + 5305*w + 7150)

inf --> 1 --> 0 --> -11/25 --> -22/17 --> 22/5 --> 187/70 --> -11/25

The first line gives the formula for the function; the second gives the forward orbit of infinity (inf). Note the repetition of -11/25 in this example, showing the cycle of period 4.

Here is a sample entry from the Small file:

phi(w) = (-60*w^2 + 24*w + 36)/(-60*w^2 + 143*w - 6)

inf --> 1 --> 0 --> -6 --> 3/4 --> 3/10 --> 6/5 --> -3/11 --> -48/95 --> -78/853

hhat(inf) = 0.011308589139336591803864543246008924166

h(phi) = 12.056347989725183163302238437525483561

hhat(inf)/h(phi) = 0.00093797799706628772830489630836933015025

Again, the first line gives a formula for the function, and the second gives the first 10 iterates in the (non-preperiodic) forward orbit of infinity (inf).

The third line gives an approximation of the canonical height of infinity; the fourth gives the height of phi as a point in the moduli space; and the fifth is the quotient of these two heights.

NOTE: the decimal approximation for hhat(inf), and hence also the quotient hhat(inf)/h(phi), are only reliable to four or five places past the decimal point. They are simply the raw outputs from PARI, which is why so many digits appear in the expansion, even though most of the digits that appear are completely unreliable. However, the first four and usually five digits after the decimal point are accurate.