next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
SymbolicPowers :: SymbolicPowers

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) = ∩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,

Contributors

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

A Quick Introduction

Other Related Examples

Author

Certification a gold star

Version 2.0 of this package was accepted for publication in volume 9 of the journal The Journal of Software for Algebra and Geometry on 20 May 2019, in the article Calculations involving symbolic powers. That version can be obtained from the journal or from the Macaulay2 source code repository, http://github.com/Macaulay2/M2/blob/master/M2/Macaulay2/packages/SymbolicPowers.m2, commit number fe3eea250b0c2c9a0ebbbd84cf44b7a52da63fc0.

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.