PolynomialTraits

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PolynomialTraits_d

Definition

A model of PolynomialTraits_d is associated with a type Polynomial_d. The type Polynomial_d represents a multivariate polynomial. The number of variables is denoted as the dimension d of the polynomial, it is arbitrary but fixed for a certain model of this concept. Note that univariate polynomials are not excluded by this concept. In this case d is just set to one.

PolynomialTraits_d provides two different views on the multivariate polynomial.

The recursive view: In this view, the polynomial is considered as an element of R[x₀, … ,x_d-2][x_d-1]. That is, the polynomial is treated as a univariate polynomial over the ring R[x₀, … ,x_d-2].
The symmetric or multivariate view: This view is symmetric with respect to all variables, considering the polynomial as an element of R [x₀, … ,x_d-1].

Many functors consider the polynomial as a univariate polynomial in one variable. By default this is the outermost variable x_d-1. However, in general it is possible to select a certain variable.

Refines

AlgebraicStructureTraits

Constants

static const int

The dimension and the number of variables respectively.

Types

PolynomialTraits_d::Polynomial_d
	Type representing R[x₀, … ,x_d-1].
PolynomialTraits_d::Coefficient_type
	Type representing R[x₀, … ,x_d-2].
PolynomialTraits_d::Innermost_coefficient_type
	Type representing the base ring R.

PolynomialTraits_d::Coefficient_const_iterator
	Const iterator used to iterate through all coefficients of the polynomial.

PolynomialTraits_d::Innermost_coefficient_const_iterator
	Const iterator used to iterate through all innermost coefficients of the polynomial.

template <typename T, int d>
PolynomialTraits_d:: struct Rebind
	This nested template class has to define a type Other which is a model of the concept PolynomialTraits_d, where d is the number of variables and T the Innermost_coefficient_type.

Functors

In case a functor is not provided it is set to CGAL::Null_functor.

PolynomialTraits_d::Construct_polynomial
	A model of PolynomialTraits_d::ConstructPolynomial.

PolynomialTraits_d::Get_coefficient
	A model of PolynomialTraits_d::GetCoefficient.

PolynomialTraits_d::Get_innermost_coefficient
	A model of PolynomialTraits_d::GetInnermostCoefficient.

PolynomialTraits_d::Construct_coefficient_const_iterator_range
	A model of PolynomialTraits_d::ConstructCoefficientConstIteratorRange.

PolynomialTraits_d::Construct_innermost_coefficient_const_iterator_range
	A model of PolynomialTraits_d::ConstructInnermostCoefficientConstIteratorRange.

PolynomialTraits_d::Swap
	A model of PolynomialTraits_d::Swap.

PolynomialTraits_d::Move
	A model of PolynomialTraits_d::Move.

PolynomialTraits_d::Degree
	A model of PolynomialTraits_d::Degree.

PolynomialTraits_d::Total_degree
	A model of PolynomialTraits_d::TotalDegree.

PolynomialTraits_d::Degree_vector
	A model of PolynomialTraits_d::DegreeVector.

PolynomialTraits_d::Leading_coefficient
	A model of PolynomialTraits_d::LeadingCoefficient.

PolynomialTraits_d::Innermost_leading_coefficient
	A model of PolynomialTraits_d::InnermostLeadingCoefficient.

PolynomialTraits_d::Canonicalize
	A model of PolynomialTraits_d::Canonicalize.

PolynomialTraits_d::Differentiate
	A model of PolynomialTraits_d::Differentiate.

PolynomialTraits_d::Evaluate
	A model of PolynomialTraits_d::Evaluate.

PolynomialTraits_d::Evaluate_homogeneous
	A model of PolynomialTraits_d::EvaluateHomogeneous.

PolynomialTraits_d::Substitute
	A model of PolynomialTraits_d::Substitute.

PolynomialTraits_d::Substitute_homogeneous
	A model of PolynomialTraits_d::SubstituteHomogeneous.

PolynomialTraits_d::Is_zero_at
	A model of PolynomialTraits_d::IsZeroAt.

PolynomialTraits_d::Is_zero_at_homogeneous
	A model of PolynomialTraits_d::IsZeroAtHomogeneous.

PolynomialTraits_d::Sign_at
	A model of PolynomialTraits_d::SignAt. In case Innermost_coefficient_type is not RealEmbeddable this is CGAL::Null_functor.

PolynomialTraits_d::Sign_at_homogeneous
	A model of PolynomialTraits_d::SignAtHomogeneous. In case Innermost_coefficient_type is not RealEmbeddable this is CGAL::Null_functor.

PolynomialTraits_d::Compare
	A model of PolynomialTraits_d::Compare. In case Innermost_coefficient_type is not RealEmbeddable this is CGAL::Null_functor.

PolynomialTraits_d::Univariate_content
	In case PolynomialTraits_d::Coefficient_type is not a model of UniqueFactorizationDomain, this is CGAL::Null_functor, otherwise this is a model of PolynomialTraits_d::UnivariateContent.

PolynomialTraits_d::Multivariate_content
	In case PolynomialTraits_d::Innermost_coefficient_type is not a model of UniqueFactorizationDomain, this is CGAL::Null_functor, otherwise this is a model of PolynomialTraits_d::MultivariateContent.

PolynomialTraits_d::Shift
	A model of PolynomialTraits_d::Shift.
PolynomialTraits_d::Negate
	A model of PolynomialTraits_d::Negate.
PolynomialTraits_d::Invert
	A model of PolynomialTraits_d::Invert.

PolynomialTraits_d::Translate
	A model of PolynomialTraits_d::Translate.
PolynomialTraits_d::Translate_homogeneous
	A model of PolynomialTraits_d::TranslateHomogeneous.

PolynomialTraits_d::Scale
	A model of PolynomialTraits_d::Scale.
PolynomialTraits_d::Scale_homogeneous
	A model of PolynomialTraits_d::ScaleHomogeneous.

PolynomialTraits_d::Make_square_free
	A model of PolynomialTraits_d::MakeSquareFree.
PolynomialTraits_d::Square_free_factorize
	In case PolynomialTraits::Polynomial_d is not a model of UniqueFactorizationDomain, this is of type CGAL::Null_functor, otherwise this is a model of PolynomialTraits_d::SquareFreeFactorize.

PolynomialTraits_d::Pseudo_division
	A model of PolynomialTraits_d::PseudoDivision.
PolynomialTraits_d::Pseudo_division_remainder
	A model of PolynomialTraits_d::PseudoDivisionRemainder.
PolynomialTraits_d::Pseudo_division_quotient
	A model of PolynomialTraits_d::PseudoDivisionQuotient.

PolynomialTraits_d::Gcd_up_to_constant_factor
	A model of PolynomialTraits_d::GcdUpToConstantFactor.
PolynomialTraits_d::Integral_division_up_to_constant_factor
	A model of PolynomialTraits_d::IntegralDivisionUpToConstantFactor.
PolynomialTraits_d::Content_up_to_constant_factor
	A model of PolynomialTraits_d::UnivariateContentUpToConstantFactor.
PolynomialTraits_d::Square_free_factorize_up_to_constant_factor
	A model of PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor.

PolynomialTraits_d::Resultant
	A model of PolynomialTraits_d::Resultant.

PolynomialTraits_d::Polynomial_subresultants
	Either CGAL::Null_functor or a model of PolynomialTraits_d::PolynomialSubresultants.

PolynomialTraits_d::Polynomial_subresultants_with_cofactors
	Either CGAL::Null_functor or a model of PolynomialTraits_d::PolynomialSubresultants_with_cofactors.

PolynomialTraits_d::Principal_subresultants
	Either CGAL::Null_functor or a model of PolynomialTraits_d::PrincipalSubresultants.

PolynomialTraits_d::Sturm_habicht_sequence
	Either CGAL::Null_functor or a model of PolynomialTraits_d::SturmHabichtSequence.

PolynomialTraits_d::Sturm_habicht_sequence_with_cofactors
	Either CGAL::Null_functor or a model of PolynomialTraits_d::SturmHabichtSequenceWithCofactors.

PolynomialTraits_d::Principal_sturm_habicht_sequence
	Either CGAL::Null_functor or a model of PolynomialTraits_d::PrincipalSturmHabichtSequence.

Has Models

CGAL::Polynomial_traits_d<Polynomial_d>

Next: PolynomialTraits_d::ConstructPolynomial

Navigation: Up Table of Contents Package Overview Bibliography Index Title Acknowledging CGAL

CGAL Open Source Project. Release 3.6.1. 29 June 2010.