# MultiplicitySequence -- multiplicity sequence of ideals

## Description

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:

• [AM97] Achilles-Manaresi, Multiplicities of a bigraded ring and intersection theory. Math. Ann. 309, 573–591 (1997).
• [AR01] Achilles-Rams, Intersection numbers, Segre numbers and generalized Samuel multiplicities. Arch. Math. (Basel) 77, 391–398 (2001)
• [JM13] Jeffries-Montaño, The j-multiplicity of monomial ideals, Math. Res. Lett. 20 (2013), no. 4, 729–744.
• [NU10] Nishida-Ulrich, Computing j-multiplicities, J. Pure Appl. Algebra, 214(12) (2010), 2101–2110.
• [PTUV20] Polini-Trung-Ulrich-Validashti, Multiplicity sequence and integral dependence. Math. Ann. 378 (2020), no. 3-4, 951–969.
• [SH06] Swanson-Huneke, Integral Closure of Ideals, Rings, and Modules, London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006).

## Authors

• Justin Chen
• Youngsu Kim
• Jonathan Montaño

## Version

This documentation describes version 0.7 of MultiplicitySequence.

## Source code

The source code from which this documentation is derived is in the file MultiplicitySequence.m2.

## Exports

• Functions and commands
• Methods
• "grGr(Ideal)" -- see grGr -- the bigraded ring Gr_m(Gr_I(R))
• "grGr(Ideal,Ideal)" -- see grGr -- the bigraded ring Gr_m(Gr_I(R))
• "hilbertSequence(Ideal)" -- see hilbertSequence -- the Hilbert sequence of a multi-graded module
• "hilbertSequence(Module)" -- see hilbertSequence -- the Hilbert sequence of a multi-graded module
• "hilbertSequence(Ring)" -- see hilbertSequence -- the Hilbert sequence of a multi-graded module
• "jMult(Ideal)" -- see jMult -- the j-multiplicity of an ideal