Binomials -- a package for binomial ideals

Description

Binomials is a package for binomial ideals with a particular focus on intersection decompositions and associated primes. For instance, if the input is a unital binomial ideal (that is generated by monomials and differences of monomials) then the functionbinomialPrimaryDecompositioncomputes a primary decomposition into binomial ideals. To this end a cyclotomic field extension of the coefficient field may be necessary which is automatically constructed using the package Cyclotomic. Binomials also implements the data type partialCharacter (see [ES96]) and several convenience functions to transform binomials into exponent vectors and vice versa. Those may be useful for manual inspection of binomial ideals.There is no special datatype for binomial ideals implemented, one just uses ideals.

Literature

[ES96] Binomial ideals (D. Eisenbud, B.Sturmfels, 1996).
[DMM10] Combinatorics of binomial primary decomposition (A. Dickenstein, L. Matusevich, E.Miller, 2010)
[OS00] Cellular Binomial Ideals. Primary Decomposition of Binomial Ideals (I. Ojeda, R. Piedra-Sanchez, 2000)
[Alt00] The chain property for the associated primes of A-graded ideals (K. Altmann, 2000)
[KM11] Decompositions of commutative monoid congruences and binomial ideals (T. Kahle, E. Miller, 2011)

Author

Thomas Kahle <[email protected]>

Certification

Version 1.0 of this package was accepted for publication in volume 4 of The Journal of Software for Algebra and Geometry: Macaulay2 on 2012-02-06, in the article Decompositions of binomial ideals (DOI: 10.2140/jsag.2012.4.1). That version can be obtained from the journal.

Version

This documentation describes version 1.2.1 of Binomials.

Citation

If you have used this package in your research, please cite it as follows:

@misc{BinomialsSource,
  title = {{Binomials: specialized routines for binomial ideals. Version~1.2.1}},
  author = {Thomas Kahle},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

@article{BinomialsArticle,
  title = {{Decompositions of binomial ideals}},
  author = {Thomas Kahle},
  journal = {The Journal of Software for Algebra and Geometry: Macaulay2},
  volume = {4},
  year = {2012},
}

Exports

Types
- PartialCharacter -- the class of all partial characters
Functions and commands
- BCD -- Binomial Cellular Decomposition
- binomialAssociatedPrimes -- Associated primes of a binomial ideal
- binomialCellularDecomposition -- Binomial Cellular Decomposition
- binomialIsPrimary -- test for primary binomial ideals
- binomialIsPrime -- test for primeness of a binomial ideal
- binomialMinimalPrimes -- minimal primes of a binomial Ideal
- binomialPrimaryDecomposition -- Binomial Primary Decomposition
- binomialRadical -- Radical of a binomial ideal
- binomialSolve -- solving zero-dimensional binomial Ideals
- binomialUnmixedDecomposition -- Binomial Unmixed Decomposition
- BPD -- Binomial Primary Decomposition
- BUD -- Binomial Unmixed Decomposition
- cellularBinomialAssociatedPrimes -- Associated primes of a cellular binomial ideal
- cellularBinomialIsPrimary -- test for primaryness of a binomial ideal
- cellularBinomialPrimaryDecomposition -- Primary decomposition of a cellular binomial ideal
- cellularBinomialRadical -- Radical of a cellular binomial ideal
- cellularBinomialUnmixedDecomposition -- Unmixed decomposition of a cellular binomial ideal
- extractInclusionMinimalIdeals -- Extract inclusion minimal ideals from a list of ideals
- idealFromCharacter -- Generate a lattice ideal from a character.
- isBinomial -- testing for unital binomial ideals
- isUnital -- see isBinomial -- testing for unital binomial ideals
- isCellular -- testing for cellular binomial ideals
- latticeBasisIdeal -- construct the ideal whose generators correspond to generators of an integer lattice
- makeBinomial -- make a binomial from an exponent vector and a coefficient
- partialCharacter -- Computing the partial character of a cellular binomial ideal
- randomBinomialIdeal -- Random Binomial Ideals
Methods
- binomialCellularDecomposition(Ideal) -- see binomialCellularDecomposition -- Binomial Cellular Decomposition
- binomialIsPrime(Ideal) -- see binomialIsPrime -- test for primeness of a binomial ideal
- binomialMinimalPrimes(Ideal) -- see binomialMinimalPrimes -- minimal primes of a binomial Ideal
- binomialPrimaryDecomposition(Ideal) -- see binomialPrimaryDecomposition -- Binomial Primary Decomposition
- binomialUnmixedDecomposition(Ideal) -- see binomialUnmixedDecomposition -- Binomial Unmixed Decomposition
- cellularBinomialAssociatedPrimes(Ideal) -- see cellularBinomialAssociatedPrimes -- Associated primes of a cellular binomial ideal
- cellularBinomialIsPrimary(Ideal) -- see cellularBinomialIsPrimary -- test for primaryness of a binomial ideal
- cellularBinomialPrimaryDecomposition(Ideal) -- see cellularBinomialPrimaryDecomposition -- Primary decomposition of a cellular binomial ideal
- cellularBinomialRadical(Ideal) -- see cellularBinomialRadical -- Radical of a cellular binomial ideal
- cellularBinomialUnmixedDecomposition(Ideal) -- see cellularBinomialUnmixedDecomposition -- Unmixed decomposition of a cellular binomial ideal
- extractInclusionMinimalIdeals(List) -- see extractInclusionMinimalIdeals -- Extract inclusion minimal ideals from a list of ideals
- idealFromCharacter(Ring,PartialCharacter) -- see idealFromCharacter -- Generate a lattice ideal from a character.
- isCellular(Ideal) -- see isCellular -- testing for cellular binomial ideals
- partialCharacter(Ideal) -- see partialCharacter -- Computing the partial character of a cellular binomial ideal
Symbols
- CellVariables -- cellular variables
- ReturnCellVars -- return the cellular variables
- ReturnPChars -- return two partial characters
- ReturnPrimes -- return two associated primes

For the programmer

The object Binomials is a package, defined in Binomials.m2.

The source of this document is in Binomials.m2:1403:0.