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SymbolicPowers :: SymbolicPowers

SymbolicPowers -- A package for computing 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) = ∩p ∈Ass(R/I)(InRp ∩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) = ∩p ∈Min(R/I)(InRp ∩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

Other Related Examples

Author

Version

This documentation describes version 2.0 of SymbolicPowers.

Source code

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

Exports

  • Functions and commands
  • Symbols
    • CIPrimes -- an option to compute the symbolic power by taking the intersection of the powers of the primary components
    • InSymbolic -- an optional parameter used in containmentProblem.
    • SampleSize -- optional parameter used for approximating asymptotic invariants that are defined as limits.
    • UseMinimalPrimes -- an option to only use minimal primes to calculate symbolic powers
    • UseWaldschmidt -- optional input for computing a lower bound for the resurgence of a given ideal.