The signature of a monomial ideal

Abstract

The irreducible decomposition of a monomial ideal has played an important role in combinatorial commutative algebra, with applications beyond pure mathematics, such as biology. Given a monomial ideal $ I $ of a polynomial ring $ S = k[{\bf x}] $ over a field $ k $ and variables $ {\bf x} = \{x_1, \ldots, x_n\} $, its incidence matrix, is the matrix whose rows are indexed by the variables $ {\bf x} $ and whose columns are indexed by its minimal generators. The main contribution of this paper is the introduction of a novel invariant of a monomial ideal $ I $, termed its signature, which could be thought of as a type of canonical form of its incidence matrix, and the proof that two monomial ideals with the same signature have essentially the same irreducible decomposition.

Keywords

• irreducible decomposition
• monomial ideal
• signature
• incidence matrix
• regular sequences
• minimal free resolution