Classifying (near)-Belyi maps with Five Exceptional Points
Accompanying data/programs/examples/computations for the paper
Mark van Hoeij and Vijay Kunwar.
- Files labelled "Input" are self-contained and can be run directly in Maple (with Maple's read command, or with copy/paste).
- Files labelled "Input*" require 1 more file to run: AllPrograms which contains programs from "Input" files.
- Files labelled "Input**" require the file AllData as well, which contains the data from "Input" files for Sections 1 - 5.4.
- Section 1:
- Program+example for (k,l,m)-exceptional points: Input and output.
- Section 2:
- An example of computing a rational function with a prescribed branching pattern:
Input and output.
- Equation (2) and Definition 2.2: Input and output.
- Section 2.1:
- Section 3.1:
- From f to a k-constellation, and from f to a plot of its dessin d'enfant:
Input and pdf output.
- Section 4.1:
- Algorithm Compute3Constellations of degree <= N: Input and output.
- Section 4.2:
- Section 4.3:
- Algorithm UniqueRepresentative, implementation and example from Section 3.1: Input and output.
- Section 4.4:
- Algorithm PlanarDessins, run with inputs: (k,2,infinity)-(count or Count) = d, with k in {3,4,6} and d in {3,4,5}.
Input and output.
- Completeness proof for the table of genus 0 Belyi functions with (3,2,infinity)-Count 5.
- Part 1: Compute a 3-constellation [g0,g1] (g_infinity is discarded) for each Belyi function.
Input and output.
- Part 2: Compare with the output of PlanarDessins([3,2,infinity],5,"Count").
Input and output.
- Completeness proof for the table of genus 0 Belyi functions with (4,2,infinity)-Count 5.
Input* and output.
- Completeness proof for the table of genus 0 Belyi functions with (6,2,infinity)-Count 5.
Input* and output.
- Section 5.2:
- Sections 5.3, 5.4 and 5.5:
- Example how to reduce an Belyi-1 map to a duplicate-free Belyi-1 map: Input
and output.
- Check that all our Belyi-1 maps are gap-free and duplicate-free.
Input* and output.
- Completeness proof for our table of genus 0 Belyi-1 functions.
Input** and output.
- Section 5.6:
- Completeness proof for Belyi maps with count = 5 and Count > 5. Input** and output.
- Section 6:
- Section 7:
- Five-point invariants I5 and I5tilde. Implementation (contains an explanation and an example).
- Remark 7.1: Another way to prove completeness of the tables of Belyi and Belyi-1 functions:
- Goal (c): FindF. This file needs the files AllData and AllPrograms.
When first used, FindF also needs to read ComputeInvariants
(which computes+stores invariants, and is then no longer needed). test file.
- Section 8: FiveSingSolver and test file.
- Concluding remarks:
- Belyi maps for d=4 have been tabulated
for differential equations with
or without logarithmic singularities.
- To do: Implement FourSingSolver and combine with FiveSingSolver to handle d <= 5.
- We tabulated dessins up to d=7, here is a list up to d=6 with group and decomposition data.
However, it would be impractical to build a table of Belyi functions for d=6.