multiIdeal< K > Class Template Reference

#include <multigrobner.H>

List of all members.

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Detailed Description

template<class K>
class multiIdeal< K >

multiIdeal class

We think of this as a submodule of a free module, over a polynomial ring, given by a set of generators. The methods are analogous to those of the Ideal class, indeed the code is almost a copy & paste.

Definition at line 128 of file multigrobner.H.

Public Member Functions
void	grobnerize_with_coefficients (Matrix< Polynomial< K > > &M)
	Buchberger's algorithm.
void	syzygy (Matrix< Polynomial< K > > &M) const
	computes generators for the module of syzygies
Public Attributes
list< multiPolynomial< K > >	generators
	a collection of polynomial which generate the ideal
Private Member Functions
void	insert (const multiPolynomial< K > &mP, const Matrix< Polynomial< K > > &coords)
long	simplify ()
	gets rid of elements in the grobner basis, so it becomes minimal
Private Attributes
list< Matrix< Polynomial< K > > >	coeffs
	used by grobnerize_with_coefficients

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Member Function Documentation

template<class K> void multiIdeal< K >::grobnerize_with_coefficients (  Matrix< Polynomial< K > > &  M  )  [inline]

Buchberger's algorithm. This adds elements in the family of generators so that it becomes a grobner basis. This is well-suited for the computation of syzygies: a matrix T is filled such that G=T.F, where F resp G, is the old, resp the new collection of generators, viewed as matrices (each row corresponding to a generator); and moreover the fact that G "contains" F makes the computation of syzygies straightforward. See also: syzygy() Definition at line 224 of file multigrobner.H. References Polynomial< K >::alphabet, Matrix< T >::get_data_from_matrix_list(), multiPolynomial< K >::is_zero(), and Table< T >::resize(). Referenced by DerivationAnalyzer::constant_subalgebra().

template<class K> long multiIdeal< K >::simplify (   )  [inline, private]

gets rid of elements in the grobner basis, so it becomes minimal Definition at line 162 of file multigrobner.H. References Table< T >::ncols().

template<class K> void multiIdeal< K >::syzygy (  Matrix< Polynomial< K > > &  M  )  const [inline]

computes generators for the module of syzygies This computes a matrix M whose rows are generators for the module of syzygies of the generators defining this multiideal. These will be highly redundant (typically some rows are even 0). Important: It is assumed that the generators of the ideal are in grobner form. If they are not, luckily, it is still easy to compute the syzygies: call grobnerize_with_coefficients and obtain in this way the matrix expressing the new generators -- call it T. Then compute M using syzygy(). The syzygies for the original generators are given by M*T. Definition at line 376 of file multigrobner.H. References Matrix< T >::get_data_from_matrix_list(). Referenced by DerivationAnalyzer::constant_subalgebra().

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Member Data Documentation

template<class K> list< Matrix< Polynomial<K> > > multiIdeal< K >::coeffs [private]

used by grobnerize_with_coefficients This holds the coefficients of the generators which are being computed by grobnerize_with_coefficients; they allow you to express the new generators in terms of the "old" ones, that is, the generators which existed before you called grobnerize_with_coefficients. If there were n of them,then each matrix in this list has 1 line and n columns. It starts as "the identity matrix". Definition at line 140 of file multigrobner.H.

template<class K> list< multiPolynomial<K> > multiIdeal< K >::generators

a collection of polynomial which generate the ideal Definition at line 206 of file multigrobner.H. Referenced by DerivationAnalyzer::constant_subalgebra().

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The documentation for this class was generated from the following file:

• include/multigrobner.H

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