This page contains electronic data for Alex Molnar's M.Sc. thesis, "Fractional Linear Minimal Models of Rational Functions."

Note that for a sequence [c[0],...,c[2n+1]] of distinct rational numbers, there is a unique rational function f of degree at most n such that f(c[i])=c[i+1] for i = 0,...,2n.

For n=2, we determined all such sequences with c[0]=0, c[1] in {1,...,100}, c[i] in {-100,...,100} for i = 2,...,5 such that the corresponding rational function f is
*minimal* (see thesis for definition) and such that
c[6]=f(c[5]) is an integer as well. There are 2190 such sequences. The corresponding functions f are minimal rational functions of degree 2 with at least 7 integers in an orbit. The results are
available in this file:

Among these, the following examples have c[7]=f(c[6]) an integer as well. These are minimal rational functions of degree 2 with 8 integers in an orbit. This is the highest known value for number of integers in an orbit of a minimal rational function of degree 2.

[0, 1, 4, 11, 12, 7,15,-374] [0, 7, -8, -21, -5, -33,-26,-1020] [0, 9, -10, 2, 12, -5,1,10] [0, 35, 27, 17, 18, 21,26,-99]

We also found one example where an orbit contains an integer after a non-integral value

[0, 1, 2, 3, 7, 5,4,41/13,-40]

For n=3, we determined all such sequences with c[0]=0, c[1] in {1,...,10}, c[i] in {-10,...,10} for i = 2,...,7 such that the corresponding rational function f is minimal and such that c[8]=f(c[7]) is an integer as well. There are 6508 such sequences. The corresponding functions f are minimal rational functions of degree 3 with at least 9 integers in an orbit. The results are available in this file:

Among these, the following examples have c[9]=f(c[8]) an integer as well. These are minimal rational functions of degree 3 with 10 integers in an orbit. This is the highest known value for number of integers in an orbit of a minimal rational function of degree 3.

[0, 1, -3, -4, -1, -2, -6, 8, -11, -582] [0, 1, -1, 7, -5, -3, -8, -7, -2, -37] [0, 1, 2, -2, -10, -8, -7, -6, -4, -83] [0, 1, 9, -3, -5, -9, -4, -6, 2, 18] [0, 2, -6, 6, -3, 3, -9, 5, -5, 8] [0, 2, -6, 8, -2, 1, -1, 5, 15, -67] [0, 2, -5, 5, -1, 1, -7, 7, 25, 87] [0, 2, -3, 1, -8, -2, 3, -1, 12, 80] [0, 2, -3, 3, 1, -9, -1, 6, 11, 321] [0, 2, -2, -6, -5, -3, 3, 1, 9, 5] [0, 2, -1, 3, -6, -5, -8, -2, 4, 244] [0, 2, 1, 4, 8, 7, 6, -1, -2, -13] [0, 2, 4, 1, 3, -5, 7, 9, 6, -92] [0, 2, 6, 3, 10, 7, -5, -8, -18, 2735] [0, 3, -10, 8, -7, 7, -1, 5, 13, 89] [0, 3, -6, 9, 4, -1, -2, -3, 2, -83] [0, 4, -3, -2, 3, -1, 9, -8, -12, -13] [0, 4, -2, 3, 1, 5, -4, 10, -7, -24] [0, 4, -2, 3, 2, -1, 6, -4, -22, -13] [0, 4, -2, 6, 1, 3, 7, -1, 5, -421] [0, 4, 10, -1, 5, 9, -5, 1, 3, 41] [0, 6, 3, -1, 5, -4, 8, 2, -6, -5] [0, 7, -4, 5, 6, 10, 3, 4, 1, -180] [0, 7, 4, -5, 2, -3, 9, 1, -2, 265] [0, 8, -5, 3, -2, 9, -4, 7, -6, -539] [0, 8, 2, 3, -1, 5, -2, 7, 1, 6] [0, 9, -7, 5, -10, -1, -2, -9, 1, -969] [0, 9, 6, 7, 4, 10, -2, -5, 40, 37]

Another way to find orbits of length 10 is to start with an orbit of length 9, construct the corresponding rational map F/G, and search for integral roots of F. We found 25 maps with 10 integral points in the orbit of 0 this way. However 11 were PGL_{2}(**Z**) conjugate to the maps above, so this gives 14 distinct orbits, listed below, with at least 10 integers.

[0, 2, -5, 11, 3, -1, 1, 4, -4, 328] [0, 4, 14, -4, 2, 5, -1, 3, 7, 35] [0, 5, 3, 6, 4, 13, 11, 12, 10, -3] [0, 5, 10, 9, 14, 7, 8, 4, 6, 11] [0, 6, 11, 2, 5, 16, 7, 10, 9, 18] [0, 7, 14, 5, 2, 4, -1, 6, 1, -34] [0, 8, 7, 3, 9, 11, 5, 6, 10, 2] [0, 9, 6, 14, 18, 12, 3, 7, 13, -1148] [0, 9, 15, 10, 18, 6, 16, 8, 11, 20] [0, 11, 10, 5, 14, 17, 12, 7, 6, -87] [0, 12, 8, 16, 20, 10, 13, 6, 15, -12] [0, 13, 15, 8, 10, 18, 14, 16, 9, -19] [0, 17, 13, 19, 20, 14, 21, 15, 16, -15] [0, 20, 15, 17, 21, 14, 16, 18, 11, -255]

We also found some examples where an orbit contains an integer after a non-integral value

[0, 1, -1, -9, -5, -4, -3, 3,∞,-6] [0, 1, 8, 5, 4, 3, 2, -2,∞,7] [0, 2, 5, -3, 9, -2, 7, 1,∞,-24] [0, 6, 1, 3, 7, -1, 8, 2,∞,-35] [0, 7, 3, 9, 10, 5, 8, 4, 20, 28/5, -160] [0, 9, -10, -4, 3, 8, 5, 10,∞,-157]Since submission, we have noticed some errors that were not caught. The following will contain an updated list of errors brought to my attention:

> Attach("ratmin.m"); > phi:=interpolate([0,1,4,11,12,7,15,-374]); > K:=Parent(phi); > phi; (86*z^2 - 1068*z - 338)/(z^2 + 7*z - 338) > // Test that phi is indeed (affine) minimal: > IsAffineMinimal(phi); true (86*z^2 - 1068*z - 338)/(z^2 + 7*z - 338) <1, 0> > // Verify that phi does not have rational preperiodic points > RationalPreperiodicPoints(phi); {} > // Compute the orbit of zero to verify it was interpolated correctly. > [i eq 1 select 0 else Evaluate(phi,Self(i-1)): i in [1..10]]; [ 0, 1, 4, 11, 12, 7, 15, -374, 59183/652, 23624638674/329913959 ]

sage: load AffMin.sage sage: F,G = IntOrb([0,1,4,11,12,7],2) sage: F/G (86*z^2 - 1068*z - 338)/(z^2 + 7*z - 338) sage: #Test to see whether F/G is minimal or not sage: Affine_minimal(F,G) (True, 86*z^2 - 1068*z - 338, z^2 + 7*z - 338)