Irreducible decomposition of binomial ideals

Abstract

Building on coprincipal mesoprimary decomposition [Kahle and Miller, 2014], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of binomial irreducible ideals, thus answering a question of Eisenbud and Sturmfels [1996].

15 pages, 3 figures, v2: minor changes implemented during copy-editing, final version as in Compositio Mathematica

References in corpus (1)

• Decompositions of commutative monoid congruences and binomial ideals

Cited by in corpus (8)

• Data structures for real multiparameter persistence modules
• Essential graded algebra over polynomial rings with real exponents
• Primary decomposition over partially ordered groups
• Some algebraic aspects of mesoprimary decomposition
• Binomial ideals and congruences on ${\mathbb{N}}^n$
• Mesoprimary decomposition of binomial submodules
• Standard pairs for monomial ideals in semigroup rings
• When is a polynomial ideal binomial after an ambient automorphism?