## Decompositions of binomial ideals.(English)Zbl 1440.62394

Summary: We present Binomials, a package for the computer algebra system Macaulay 2, which specializes well-known algorithms to binomial ideals. These come up frequently in algebraic statistics and commutative algebra, and it is shown that significant speedup of computations like primary decomposition is possible. While central parts of the implemented algorithms go back to a paper of Eisenbud and Sturmfels, we also discuss a new algorithm for computing the minimal primes of a binomial ideal. All decompositions make significant use of combinatorial structure found in binomial ideals, and to demonstrate the power of this approach we show how Binomialswas used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample for their conjectures.

### MSC:

 62R01 Algebraic statistics 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13F20 Polynomial rings and ideals; rings of integer-valued polynomials 62-08 Computational methods for problems pertaining to statistics

### Software:

GRIN; 4ti2; Macaulay2; Binomials.m2
Full Text:

### References:

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