The goal of this package is to compute the multiplicity sequence of an ideal $I$ in a standard graded equidimensional ring over a field $(R,m,k)$, where $m = R_+$. The multiplicity sequence is a generalization of the Hilbert-Samuel multiplicity for ideals that are not necessarily m-primary. This sequence is obtained by considering the second sum transform of the Hilbert polynomial in two variables of the bigraded ring grGr, which is the associated graded algebra of the extension of $m$ in the associated graded algebra of $I$.
The multiplicity sequence was defined by Achiles and Manaresi in intersection theory [AM97]. Its importance comes from applications to problems in singularity theory (Segre numbers [AR01]) and commutative algebra (numerical characterization of integral dependence [PTUV20, SH06]). Indeed, in [PTUV20] the authors show that in a equidimensional and universally catenary Noetherian local ring, two ideals $J\subset I$ have the same integral closure if and only if they have the same multiplicity sequence.
This package includes two different ways of computing the multiplicity sequence of an ideal. The first one uses the definition in terms of Hilbert polynomials, while the second uses a general element approach based on [AM97] (see also [PTUV20]). The package also contains a method that computes all of the coefficients of the Hilbert polynomial of a multi-graded module. These numbers can be seen as the generalizations of Hilbert coefficients for ideals that are not necessarily m-primary.
One of the terms of the multiplicity sequence is the j-multiplicity, another important invariant of an ideal in multiplicity theory. This package also contains a method jMult which computes the j-multiplicity of an ideal using Theorem 3.6 in [NU10], based on code written by H. Schenck and J. Validashti. There is also a method monjMult which computes the j-multiplicity of a monomial ideal via polyhedral volume computations, using a result of [JM13]. The package also includes several functions related to integral dependence of monomial ideals, such as Newton polyhedron, analytic spread, and monomial reductions.
The second author thanks D. Eisenbud, D. Grayson, and M. Stillman for organizing a Macaulay2 day during the special year in commutative algebra 2012-2013 at MSRI where he learned how to write a package.
References:
Version 0.7 of this package was accepted for publication in volume 14 of Journal of Software for Algebra and Geometry on 2023-10-31, in the article Computing multiplicity sequences (DOI: 10.2140/jsag.2024.14.13). That version can be obtained from the journal or from the Macaulay2 source code repository.
This documentation describes version 0.7 of MultiplicitySequence.
The source code from which this documentation is derived is in the file MultiplicitySequence.m2.
The object MultiplicitySequence is a package.