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InvolutiveBases :: InvolutiveBases

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 Gröbner 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 resolution, see Involutive.

Some references:

  • J. Apel, The theory of involutive divisions and an application to Hilbert function computations. J. Symb. Comp. 25(6), 1998, pp. 683-704.
  • V. P. Gerdt, Involutive Algorithms for Computing Gröbner Bases. In: Cojocaru, S. and Pfister, G. and Ufnarovski, V. (eds.), Computational Commutative and Non-Commutative Algebraic Geometry, NATO Science Series, IOS Press, pp. 199-225.
  • V. P. Gerdt and Y. A. Blinkov, Involutive bases of polynomial ideals. Minimal involutive bases. Mathematics and Computers in Simulation 45, 1998, pp. 519-541 resp. 543-560.
  • M. Janet, Lecons sur les systemes des equationes aux derivees partielles. Cahiers Scientifiques IV. Gauthiers-Villars, Paris, 1929.
  • J.-F. Pommaret, Partial Differential Equations and Group Theory. Kluwer Academic Publishers, 1994.
  • W. Plesken and D. Robertz, Janet's approach to presentations and resolutions for polynomials and linear pdes. Archiv der Mathematik 84(1), 2005, pp. 22-37.
  • D. Robertz, Janet Bases and Applications. In: Rosenkranz, M. and Wang, D. (eds.), Gröbner Bases in Symbolic Analysis, Radon Series on Computational and Applied Mathematics 2, de Gruyter, 2007, pp. 139-168.
  • W. M. Seiler, A Combinatorial Approach to Involution and delta-Regularity: I. Involutive Bases in Polynomial Algebras of Solvable Type. II. Structure Analysis of Polynomial Modules with Pommaret Bases. Preprints, arXiv:math/0208247 and arXiv:math/0208250.

Author

Version

This documentation describes version 1.10 of InvolutiveBases.

Source code

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

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(InvolutiveBasis), 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
    • janetResolution(InvolutiveBasis), see janetResolution -- construct a free resolution for a given ideal or module using Janet bases
    • 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)
  • 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