Programs/examples/computations needed for the completeness proof for the table and 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*" read other file(s) listed on this webpage. Unpack the file FiveSingFiles.zip and move the files where Maple's "read" command can find them.

• 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:
  • Number of exceptional points/branchings: Input and output (for both definitions, referred to as "count" and "Count").
  • Enumerate planar branching patterns: Input and output (using "Count").
• 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:
  • Weighted count example: Input and output (for both definitions).
• 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.
    Remark: We improved algorithm PlanarDessins and now have an efficient algorithm to FindDessins for arbitrary branch patterns.

  • 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.
  • Same computation for Count instead of count.
• Section 5.2:
  • Build Tables 2 and 3. Input and output.
• 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:
  • Verification that column Belyi ∈ Belyi⁽¹⁾ has all Belyi maps with count = 5 and Count > 5. Input* and output.
• Section 6:
  • Compute Belyi⁽²⁾ maps. Implementation.
• Section 7:
  • Five-point invariants I5 and I5tilde. Implementation (contains an explanation and an example).
  • Section 7 + Section 4.4: Another way to prove completeness of the tables of Belyi and Belyi⁽¹⁾ functions:
    • Input* and output which lists the number of non-equivalent entries (in the Belyi case, it lists that number for each degree separately).
    • Belyi completeness: The degree lists match those in first output file from section 4.4 (an independent computation that used only combinatorial methods).
    • Belyi⁽¹⁾ completeness: Compare the number of entries with section 5.2 (which computed orbits N_{1..65} N{66..68} O_{1..20} and P_{1..12} using only combinatorial methods).
  • Goal (c): FindF and input* (test file).
• Section 8: FiveSingSolver and input* (test file).
• Concluding remark: Belyi maps for d=4 have been tabulated, parametric cases or non-parametric cases. Those tables can be used to implement a FourSingSolver.