# Integers in orbits of rational funcions

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.

## Minimal rational functions of degree 2

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]
```

## Minimal rational functions of degree 3

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 PGL2(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:

## Software

ratmin.m - Magma software to compute minimal models and preperiodic points of rational functions over Q. The code is largely self-explanatory. Below we show a little example session that illustrates typical use.

```> 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 ]
```

AffMin.sage - Sage software to compute minimal models over Q. Below we show an example of its use.

```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)
```