SymbolicPowers -- symbolic powers of ideals

Description

This package gives the ability to compute symbolic powers, and related invariants, of ideals in a polynomial ring or a quotient of a polynomial ring. For example, in the context of the default behavior, symbolicPower assumes the following definition of the symbolic power of an ideal $I$, $$I^{(n)} = \cap_{p \in Ass(R/I)}(I^nR_p \cap R ),$$ as defined by M. Hochster and C. Huneke.

Alternatively, as defined in Villarreal, symbolicPower has the option to restrict to minimal primes versus use all associated primes with UseMinimalPrimes. In particular, the symbolic power of an ideal $I$ is defined as $$I^{(n)} = \cap_{p \in Min(R/I)}(I^nR_p \cap R ),$$ where $Min(R/I)$ is the set of minimal primes in $I$,

• M. Hochster and C. Huneke. Comparison of symbolic and ordinary powers of ideals. Invent. Math. 147 (2002), no. 2, 349–369.
• R. Villarreal. Monomial algebras. Second edition. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton, FL, 2015. xviii+686 pp. ISBN: 978-1-4822-3469-5.
• Hailong Dao, Alessandro De Stefani, Eloísa Grifo, Craig Huneke, and Luis Núñez-Betancourt. Symbolic powers of ideals.in Singularities and foliations. Geometry, topology and applications, pp. 387-432, Springer Proc. Math. Stat., 222, Springer, Cham, 2018. See https://arxiv.org/abs/1708.03010.

Contributors

The following people have generously contributed code or worked on our code at various Macaulay2 workshops.

• Ben Drabkin
• Andrew Conner
• Alexandra Seceleanu
• Branden Stone
• Xuehua (Diana) Zhong

A Quick Introduction

• Computing symbolic powers of an ideal
• Alternative algorithm to compute the symbolic powers of a prime ideal in positive characteristic

Other Related Examples

• The Containment Problem
• Sullivant's algorithm for primes in a polynomial ring
• The Packing Problem