GradedAlgebra< K > Class Template Reference

#include <gradedalgebras.H>

Inheritance diagram for GradedAlgebra< K >:

List of all members.

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Detailed Description

template<class K>
class GradedAlgebra< K >

GradedAlgebra class: an affine algebra with a grading.

All the methods are obvious extensions of those of Affinealgebra.

Definition at line 14 of file gradedalgebras.H.

Public Member Functions
Polynomial< K >	new_variable (const string &name, long degree)
virtual void	swap_variables_order (long i, long j, list< Polynomial< K > > &polys)
	changes the order of two variables
virtual void	swap_variables_order (long i, long j)
virtual void	kill_top_variable (const Polynomial< K > &replacement, list< Polynomial< K > > &polys)
	replaces the top variable by a polynomial in the others
virtual void	kill_top_variable (const Polynomial< K > &replacement)
long	hom_degree_of (const Polynomial< K > &x) const
	homogeneous degree of an element
bool	is_homogeneous (const Polynomial< K > x) const
Polynomial< K >	hom_part_of (const Polynomial< K > &x, long n) const
void	add_relation (const Polynomial< K > &rel)
	adds a generator in the relations ideal. Checks if it is 0, not if it is redundant.
void	split_and_sort_relations ()
	turns the relations into sorted, homogeneous relations.
void	even_subalgebra (GradedAlgebra< K > &ans, AffineHomomorphism< K > &inclusion) const
	the subalgebra in even degrees
long	top_degree () const
	gives a degree above which the algebra is zero
void	list_monomials (long degree, Tuple< Polynomial< K > > &result) const
	gives a K-basis of a given homogenous part
Public Attributes
map< long, long >	hom_degrees
	the homogeneous degrees
Friends
ostream &	operator<< (ostream &os, const GradedAlgebra< K > &R)
ofstream &	operator<< (ofstream &file, GradedAlgebra< K > &A)
	write to file
void	operator>> (ifstream &file, GradedAlgebra< K > &A)

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Member Function Documentation

template<class K> void GradedAlgebra< K >::add_relation (  const Polynomial< K > &  rel  )  [inline]

adds a generator in the relations ideal. Checks if it is 0, not if it is redundant. Reimplemented from AffineAlgebra< K >. Definition at line 118 of file gradedalgebras.H. Referenced by HomFinder::define_polynomial_variables(), HomFinder::extend(), HomFinder::extend_all(), HomFinder::init(), and HomFinder::test_steenrod().

template<class K> void GradedAlgebra< K >::even_subalgebra (  GradedAlgebra< K > &  ans, AffineHomomorphism< K > &  inclusion )  const [inline]

the subalgebra in even degrees This writes in 'ans' a polynomial algebra which maps onto the even-degree subalgebra of *this, via the homomorphism written in 'inclusion'. The user may or may not want to compute the kernel of 'inclusion', giving a presentation for the even-degree subalgebra. Definition at line 193 of file gradedalgebras.H. Referenced by MilnorAnalyzer::done().

template<class K> long GradedAlgebra< K >::hom_degree_of (  const Polynomial< K > &  x  )  const [inline]

homogeneous degree of an element Parameters: x:  a polynomial Returns: the homogeneous degree of the leading term of x Definition at line 75 of file gradedalgebras.H. Referenced by GradedAlgebra< F_2 >::add_relation(), DerivationAnalyzer::constant_subalgebra(), and GradedAlgebra< F_2 >::even_subalgebra().

template<class K> virtual void GradedAlgebra< K >::kill_top_variable (  const Polynomial< K > &  replacement, list< Polynomial< K > > &  polys )  [inline, virtual]

replaces the top variable by a polynomial in the others This modifies the list polys so its members take the changes into account. Reimplemented from AffineAlgebra< K >. Reimplemented in UnstableAlgebra. Definition at line 53 of file gradedalgebras.H. Referenced by Exponentiator< F_2 >::kill_extra_variables(), UnstableAlgebra::kill_top_variable(), and HomFinder::polynomial_test().

template<class K> void GradedAlgebra< K >::list_monomials (  long  degree, Tuple< Polynomial< K > > &  result )  const [inline]

gives a K-basis of a given homogenous part This writes in the Tuple 'result' a bunch of elements of the graded algebra which form a K-basis for the homogeneous part of degree 'degree'. Careful: This will only work if a Grobner basis has been computed ! in fact, what this method really does is list all powerproducts in degree 'degree' and eliminate those which reduce to 0 under reduced_form(). Definition at line 291 of file gradedalgebras.H. Referenced by HomFinder::define_polynomial_variables(), HomFinder::define_polynomial_variables_brutally(), HomFinder::init(), and GradedAlgebra< F_2 >::list_monomials().

template<class K> void GradedAlgebra< K >::split_and_sort_relations (   )  [inline]

turns the relations into sorted, homogeneous relations. This splits the current relations into homogeneous parts, and organizes the resulting homogeneous relations so that they appear in 'relations' with their degrees nondecreasing. Definition at line 141 of file gradedalgebras.H.

template<class K> virtual void GradedAlgebra< K >::swap_variables_order (  long  i, long  j, list< Polynomial< K > > &  polys )  [inline, virtual]

changes the order of two variables This corrects all the relations generators so they take the change of order into account. Likewise all polynomials created with this before should be modified. Parameters:   polys, a list of polynomials to update. Reimplemented from AffineAlgebra< K >. Reimplemented in UnstableAlgebra. Definition at line 32 of file gradedalgebras.H. Referenced by MilnorAnalyzer::re_setup(), MilnorAnalyzer::setup_for_Sq1(), and UnstableAlgebra::swap_variables_order().

template<class K> long GradedAlgebra< K >::top_degree (   )  const [inline]

gives a degree above which the algebra is zero Before using this, it is necessary to call is_finite_dimensional() (from AffineAlgebra<K>), otherwise this enters an infinite loop (and the result would make no sense of course). Also, a Grobner basis must have been computed. Note: The result given is not sharp. Definition at line 252 of file gradedalgebras.H.

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Friends And Related Function Documentation

template<class K> ofstream& operator<< (  ofstream &  file, GradedAlgebra< K > &  A )  [friend]

write to file Definition at line 379 of file gradedalgebras.H.

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Member Data Documentation

template<class K> map<long, long> GradedAlgebra< K >::hom_degrees

the homogeneous degrees Definition at line 18 of file gradedalgebras.H. Referenced by HomFinder::branch(), HomFinder::define_polynomial_variables(), HomFinder::define_polynomial_variables_brutally(), HomFinder::extend(), HomFinder::extend_all(), HomFinder::init(), MilnorAnalyzer::re_setup(), DerivationAnalyzer::setup(), MilnorAnalyzer::setup_for_Sq1(), and HomFinder::weight().

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

• include/gradedalgebras.H

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