Binomial Ideals

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  • Code Cleanup
    • Update documentation Looks good
    • make PartialCharacter a new class
        • J - variable names of the lattice
        • L - lattice
        • c - vector of coefficients
      • write the constructor based on the "partialCharacter" function
      • change the saturation function to return a list of saturated characters
      • write documentation
    • Internal documentation (Is in good shape)
    • Replace all wrong usage of "use" and other global assignment problems
    • Replace 'sub' by ring maps when applied to lists (for performance)
    • Implement the new type BinomialIdeal
  • Implement Tests
    • Migrate existing tests from benchmark.m2
    • Tests with Random ideals?
      • How to generate a random binomial (actually pure-difference) ideal ?
  • Performance Ideas
    • Axis saturation should be implemented in C? (better Gröbner walks)
    • Mapping precomputed Gröbner bases around (better Gröbner walks)
    • Gröbner Walks
      • Need to look at Mike's experimental code
      • Anders' gfan is probably not useful here
    • Handling of many std-monomials
      • Idea: do an irreducible decomposition and then use 'box-search' (probably not useful)
      • Randomized search for associated lattices (implemented, leads to good speed-ups in some examples)

Here are timings of my 4 favorite examples:

  179.673 seconds
  4.18678 seconds
  139.758 seconds
  2.57416 seconds
 73.0106 seconds
 2.84043 seconds
 136.272 seconds
 2.12938 seconds
  • Decomposing Meyr-Meyer Ideals
    • (2,1) can be done now.
    • (2,2) needs time and a machine with > 20GB RAM.
      • 88 of 89 cellular components are done, last one is out of reach for now.
  • Long Term
    • Finite Fields support
    • Parallelism (tried - needs work in the core)

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