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InvolutiveBases -- Methods for Janet bases and Pommaret bases in Macaulay 2

Description

InvolutiveBases is a package which provides routines for dealing with Janet and Pommaret bases.

Janet bases can be constructed from given Gr\"obner bases. It can be checked whether a Janet basis is a Pommaret basis. Involutive reduction modulo a Janet basis can be performed. Syzygies and free resolutions can be computed using Janet bases. A convenient way to use this strategy is to use an optional argument for freeResolution, see Involutive.

Some references:

Author

Version

This documentation describes version 1.10 of InvolutiveBases.

Citation

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

@misc{InvolutiveBasesSource,
  title = {{InvolutiveBases: Methods for Janet bases and Pommaret bases in \emph{Macaulay2}. Version~1.10}},
  author = {Daniel Robertz},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Types
  • Functions and commands
    • basisElements -- extract the matrix of generators from an involutive basis or factor module basis
    • factorModuleBasis -- enumerate standard monomials
    • invNoetherNormalization -- Noether normalization
    • invReduce -- compute normal form modulo involutive basis by involutive reduction
    • invSyzygies -- compute involutive basis of syzygies
    • isPommaretBasis -- check whether or not a given Janet basis is also a Pommaret basis
    • janetBasis -- compute Janet basis for an ideal or a submodule of a free module
    • janetMultVar -- return table of multiplicative variables for given module elements as determined by Janet division
    • janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • multVar -- extract the sets of multiplicative variables for each generator (in several contexts)
    • pommaretMultVar -- return table of multiplicative variables for given module elements as determined by Pommaret division
  • Methods
    • basisElements(FactorModuleBasis) -- see basisElements -- extract the matrix of generators from an involutive basis or factor module basis
    • basisElements(InvolutiveBasis) -- see basisElements -- extract the matrix of generators from an involutive basis or factor module basis
    • factorModuleBasis(InvolutiveBasis) -- see factorModuleBasis -- enumerate standard monomials
    • invNoetherNormalization(GroebnerBasis) -- see invNoetherNormalization -- Noether normalization
    • invNoetherNormalization(Ideal) -- see invNoetherNormalization -- Noether normalization
    • invNoetherNormalization(InvolutiveBasis) -- see invNoetherNormalization -- Noether normalization
    • invNoetherNormalization(Matrix) -- see invNoetherNormalization -- Noether normalization
    • invNoetherNormalization(Module) -- see invNoetherNormalization -- Noether normalization
    • invReduce(Matrix,InvolutiveBasis) -- see invReduce -- compute normal form modulo involutive basis by involutive reduction
    • invReduce(RingElement,InvolutiveBasis) -- see invReduce -- compute normal form modulo involutive basis by involutive reduction
    • invSyzygies(InvolutiveBasis) -- see invSyzygies -- compute involutive basis of syzygies
    • isPommaretBasis(InvolutiveBasis) -- see isPommaretBasis -- check whether or not a given Janet basis is also a Pommaret basis
    • janetBasis(Complex,ZZ) -- see janetBasis -- compute Janet basis for an ideal or a submodule of a free module
    • janetBasis(GroebnerBasis) -- see janetBasis -- compute Janet basis for an ideal or a submodule of a free module
    • janetBasis(Ideal) -- see janetBasis -- compute Janet basis for an ideal or a submodule of a free module
    • janetBasis(Matrix) -- see janetBasis -- compute Janet basis for an ideal or a submodule of a free module
    • janetBasis(Module) (missing documentation)
    • janetMultVar(List) -- see janetMultVar -- return table of multiplicative variables for given module elements as determined by Janet division
    • janetMultVar(Matrix) -- see janetMultVar -- return table of multiplicative variables for given module elements as determined by Janet division
    • janetResolution(Ideal) -- see janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • janetResolution(InvolutiveBasis) -- see janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • janetResolution(Matrix) -- see janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • janetResolution(Module) -- see janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • multVar(Complex,ZZ) -- see multVar -- extract the sets of multiplicative variables for each generator (in several contexts)
    • multVar(FactorModuleBasis) -- see multVar -- extract the sets of multiplicative variables for each generator (in several contexts)
    • multVar(InvolutiveBasis) -- see multVar -- extract the sets of multiplicative variables for each generator (in several contexts)
    • pommaretMultVar(List) -- see pommaretMultVar -- return table of multiplicative variables for given module elements as determined by Pommaret division
    • pommaretMultVar(Matrix) -- see pommaretMultVar -- return table of multiplicative variables for given module elements as determined by Pommaret division
  • Symbols
    • Involutive -- compute a (usually non-minimal) resolution using involutive bases
    • multVars -- key in the cache table of a differential in a Janet resolution
    • PermuteVariables -- ensure that the last dim(I) var's are algebraically independent modulo I

For the programmer

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


The source of this document is in InvolutiveBases.m2:783:0.