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:
• 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.