Decompositions of Binomial Ideals in Macaulay 2

TL;DR

This paper presents specialized algorithms for primary decomposition of binomial ideals in characteristic zero, implemented in Macaulay 2, with recent improvements and practical examples.

Contribution

It introduces new algorithmic improvements for binomial ideal decomposition in Macaulay 2, expanding computational capabilities.

Findings

Algorithms successfully decompose binomial ideals into binomial components

Implementation handles trivial coefficient cases efficiently

Examples demonstrate practical application of the algorithms

Abstract

The package Binomials contains implementations of specialized algorithms for binomial ideals, including primary decomposition into binomial ideals. The current implementation works in characteristic zero. Primary decomposition is restricted to binomial ideals with trivial coefficients to avoid computations over the algebraic numbers. The basic ideas of the algorithms go back to Eisenbud and Sturmfels' seminal paper on the subject. Two recent improvements of the algorithms are discussed and examples are presented.