Decompositions of Binomial Ideals

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.

15 pages, Revision after referee's comments

References in corpus (1)

• On the toric algebra of graphical models

Cited by in corpus (8)

• Decompositions of Binomial Ideals in Macaulay 2
• Chordal networks of polynomial ideals
• Finding the Maximizers of the Information Divergence from an Exponential Family
• Efficiently and Effectively Recognizing Toricity of Steady State Varieties
• Eigenschemes and the Jordan canonical form
• Decompositions of Cellular Binomial Ideals
• Parametric Toricity of Steady State Varieties of Reaction Networks
• First-Order Tests for Toricity