Small Height Points for Cubic Polynomials

This page has links to all the data from the 2007 REU project with Ben Dickman, Sasha Joseph, Ben Krause, Dan Rubin, and Xinwen Zhou. The project was funded mainly by NSF Grant DMS-0600878; Krause was supported by the Amherst College Schupf Scholars program, and Zhou was supported by Amherst College Dean of Faculty student funds. For definitions and mathematical background, see our paper, available in PDF format here.
Polynomials of the form a z3 + b z + 1, with numerator and denominator of a and b less than 300:

By maximum length of
a preperiodic chain
By number of points of
small positive height
By maximum length of
a periodic cycle
By number of
preperiodic points
By total number of
small height points
   Length 6 (1)
   Length 5 (16)
   Length 4 (132)
   Length 3 (1612)
   Length 2 (197,471)
   Length 1 (282,965)
   Length 0 (161,808)
   10 points (1)
   9 points (3)
   8 points (10)
   7 points (34)
   6 points (93)
   5 points (261)
   4 points (678)
   3 points (2,546)
   2 points (5,979)
   1 point (155,757)
   0 points (478,643)
   Length 5 (1)
   Length 4 (9)
   Length 3 (101)
   Length 2 (85,950)
   Length 1 (396,136)
   Length 0 (161,808)
   10 points (1)
   9 points (14)
   8 points (24)
   7 points (118)
   6 points (198)
   5 points (403)
   4 points (1,389)
   3 points (113,430)
   2 points (84,537)
   1 point (282,083)
   0 points (161,808)
   10 points (3)
   9 points (28)
   8 points (52)
   7 points (193)
   6 points (358)
   5 points (751)
   4 points (2,314)
   3 points (115,954)
   2 points (92,221)
   1 point (432,131)

Full a z3 + b z + 1 data: all300AB1.txt.gz (12 megabyte gzip-ed file)

The master data file all300AB1.txt is a large (58 megabyte) text file with all 644,005 functions of this form that had at least one point of small height. Each column of the table above partitions the same set of 644,005 functions as specified.

Only those polynomials with at least one rational point of normalized canonical height smaller than 0.03 are listed. The remaining 11,939,398,165 such polynomials have no such points and are not listed.


Polynomials of the form a z3 + b z, with numerator and denominator of a and b less than 300:

By maximum length of
a preperiodic chain
By number of points of
small positive height
By maximum length of
a periodic cycle
By number of
preperiodic points
By total number of
small height points
   Length 4 (13)
   Length 3 (226)
   Length 2 (204,124)
   Length 1 (94,601)
   4 points (6)
   2 points (170)
   0 points (298,788)
   Length 4 (8)
   Length 2 (95,152)
   Length 1 (203,804)
   11 points (10)
   9 points (28)
   7 points (322)
   5 points (750)
   3 points (297,712)
   1 point (142)
   11 points (10)
   9 points (36)
   7 points (318)
   5 points (774)
   3 points (297,826)

Full a z3 + b z data: all300AB0.txt.gz (2.7 megabyte gzip-ed file)

The master data file all300AB0.txt is a large (23 megabyte) text file with all 298,964 functions of this form with at least one point besides x=0 of small height. Each column of the table above partitions the same set of 298,964 functions as specified.

The polynomials a z3 + b z and c2a z3 + b z are conjugate (over Q) by g(z) = cz. Thus, only one such polynomial from each Q-conjugacy class is listed. For example, z3 - z was tested, but 4z3 - z was not.

Only those polynomials with at least two rational points of normalized canonical height smaller than 0.03 are listed. The remaining 2,072,790,448 such polynomials have no small height rational points besides the fixed point at 0, and so they are not listed.


Reading the data files:

Here is a sample entry, which appears in the a z3 + b z section, under preperiodic chain of length 3, under 2 points of small positive height, under periodic cycle of length 2, under 7 preperiodic points, and under 9 total small height points.

[ 11/6 -11/6 ] [ 0 ]
-1 |-> -11/6
1 |-> 11/6
-5/6 |-> -11/6
5/6 |-> 11/6
2/3 | 0.089415001903529917240377185524395648647 | 0.019822165104976521605284672471934755346
-2/3 | 0.089415001903529917240377185524395648647 | 0.019822165104976521605284672471934755346
f(z) = -6/5*z^3 + 91/30*z
*

The final * is just a delimiter to separate entries.

The second-to-last line identifies the polynomial in question as f(z) = -(6/5)z3 + (91/30)z.

The first line lists the periodic cycles. In this case, there is a 2-cycle (11/6 mapping to -11/6 and then back to 11/6) and a fixed point (0 maps to itself).

The next few lines (if any), containing |-> arrows, list strictly preperiodic points, followed by their forward iterates until they hit a periodic or preperiodic point that has already appeared. In this case, -1 and -5/6 both map to -11/6; and 1 and 5/6 both map to 11/6.

Please note: sometimes there are more strictly preperiodic points than there are these lines of data; one line may list several such points. In addition, these lines need not end with periodic points; they may simply end with another strictly preperiodic point that has already been listed. Thus, the total number of preperiodic points is the number of points listed in the top (cycles) lines and these lines, minus the number of these preperiodic data lines.

The next few lines (if any), containing | delimiters, list points of small positive height. Each line is of the form

x | h(x) | N(x)

where x is the rational point, where h(x) is its canonical height, and N(x) is its normalized height, obtained by dividing h(x) by h+(f). Here, h(f) is the height of f (see the paper), and h+(f) is the maximum of h(f) and 1.

Please note that although about 40 decimal places are printed, only the first five places after the decimal point for N(x) are reliable.


Each data file ends with a line of the form:
TOTAL: 774
listing the total number of functions in that data file.


The master data files (all300AB1.txt and all300AB0.txt) also include, after each function, a set of lines of the form:

P 5
H 2
L 2
C 2
p 7

listing the partitioned files in which that function appears.
P is the number of preperiodic points
H is the number of points of small positive height
L is the maximum length of a periodic cycle
C is the maximum length of a preperiodic chain
p is the total number of small height points



Side Note 1: As described in the paper, the algorithm uses a search region that is guaranteed to contain all of the rational preperiodic points but which may, in rare instances, miss some points of small positive canonical height. Remark 5.2 of the paper addresses this fact, noting that such instances must be rare, and are highly unlikely to lead to a point of notably small height. Thus, while it is almost certain that a few points of normalized height smaller than 0.03 were missed by this search, it is highly unlikely that any of height smaller than, say, 0.001 were missed.


Side Note 2: In the version of the program with which this data was generated, the search region was in some rare cases slightly larger than that described in the presentation of the algorithm in section 5 of the paper. (This was done to ease the implementation of the program. Specifically, if there was a double root of f(z)=0, or no Qp-rational points, then the full disk called U0 in the paper was used as the search region at p, rather than the smaller region described in the paper.) This may have resulted in a few rare instances when a point outside the paper's search region showed up as a point of small (but still positive) canonical height in this data. Of course, that means that the search that generated this data was even more comprehensive than the search described in the paper.