Decompositions of Binomial Ideals

TL;DR

This paper introduces Binomials, a Macaulay2 package that accelerates computations on binomial ideals, including a new algorithm for minimal primes, with applications in algebraic statistics and commutative algebra.

Contribution

It presents a specialized software package with optimized algorithms for binomial ideals, including a novel minimal prime computation method.

Findings

Significant speedup in primary decomposition computations.

Successful application to compute decompositions of birth and death ideals.

Counterexample for a conjecture in algebraic statistics.

Abstract

We present Binomials, a package for the computer algebra system Macaulay2, 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 Eisenbud and Sturmfels (1996), 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 Binomials was used to compute primary decompositions of commuting birth and death ideals of Evans et al., yielding a counterexample for a conjecture therein.