Binomial Primary Decomposition

Binomials.m2 is a package for Macaulay2 to compute primary and other decompositions of binomial ideals. All algorithms are fundamentally rooted in (missing reference). The current implementation is for pure difference ideals in characteristic zero, but extensions would be possible and nice (see Todolist below!).

For more on the combinatorics of binomial primary decomposition see (missing reference) and (Kahle & Miller, 2014).

The package was first announced in Decomposing Binomial Ideals (Kahle, 2010) (arxiv:0906.4873). Later there was the paper (missing reference) in the Macaulay2 journal, which explains some of the implemented improvements.

You can send me your computational challenges! I’m always happy to collaborate on binomial ideal decomposition questions.

Usage

Binomials and Cyclotomic are shipped together with Macaulay 2. There is no need to download anything anymore.

Todolist

I think the following improvements would be very interesting. Please contact me if you want to work on them.

  • Implement the algorithms in positive characteristic.
    • This should be relatively straightforward extension of what is already there.
  • Implementation of (proper, canonical) mesoprimary decomposition as defined in (Kahle & Miller, 2014).
    • This is probably hard and there are mathematical problems to solve, e.g. how to decide if the intersection of two congruences on a monoid is trivial (= whether an intersection of 2 binomial ideals contains a binomial). This subproblem is solved in (Jensen et al., 2017)

References

  1. Decompositions of Commutative Monoid Congruences and Binomial Ideals,
    Thomas Kahle and Ezra Miller
    Algebra & Number Theory, 2014
    DOI arXiv
  2. Decompositions of Binomial Ideals,
    Thomas Kahle
    Annals of the Institute of Statistical Mathematics, 2010
    DOI arXiv
  3. Finding binomials in polynomial ideals,
    Anders Jensen, Thomas Kahle, and Lukas Katthän
    Research in the Mathematical Sciences, 2017
    DOI arXiv