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 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 Gr\"obner 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, Leçons sur les systèmes des équations aux dérivées 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.), Gr\"obner 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

Daniel Robertz <[email protected]>

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
- FactorModuleBasis -- the class of all factor module bases
- InvolutiveBasis -- the class of all involutive bases
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(ChainComplex,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
- 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(ChainComplex,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.