CGAL 6.2 - Polynomial: PolynomialTraits_d Concept Reference

CGAL 6.2 - Polynomial

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PolynomialTraits_d Concept Reference
Reference Manual » Concepts

## 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 \( R[x_0,\dots,x_{d-2}][x_{d-1}]\). That is, the polynomial is treated as a univariate polynomial over the ring \( R[x_0,\dots,x_{d-2}]\).
- The symmetric or multivariate view: This view is symmetric with respect to all variables, considering the polynomial as an element of \( R [x_0,\dots,x_{d-1}]\).

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`

See also
  `Polynomial_d`

Has models
  `CGAL::Polynomial_traits_d<Polynomial_d>`

## Concepts
concept | Canonicalize
| For a given polynomial \( p\) this `AdaptableUnaryFunction` computes the unique representative of the set. More...
concept | Compare
| This `AdaptableBinaryFunction` compares two polynomials, with respect to the lexicographic order with preference to the outermost variable. More...
concept | ConstructCoefficientConstIteratorRange
| This `AdaptableUnaryFunction` returns a const iterator range over the coefficients of the given polynomial, with respect to the outermost variable, \( x_{d-1}\). The range starts with the coefficient for \( x_{d-1}^0\). More...
concept | ConstructInnermostCoefficientConstIteratorRange
| This `AdaptableUnaryFunction` returns a const iterator range over all innermost coefficients of the given polynomial. More...
concept | ConstructPolynomial
| This `AdaptableFunctor` provides several operators to construct objects of type `PolynomialTraits_d::Polynomial_d`. More...
concept | Degree
| This `AdaptableUnaryFunction` computes the degree of a `PolynomialTraits_d::Polynomial_d` with respect to a certain variable. More...
concept | DegreeVector
| For a given `PolynomialTraits_d::Polynomial_d` \( p\) this `AdaptableUnaryFunction` returns the degree vector, that is, it returns the exponent vector of the monomial of highest order in \( p\), where the monomial order is the lexicographic order giving outer variables a higher priority. In particular, this is the monomial that belongs to the innermost leading coefficient of \( p\). More...
concept | Differentiate
| This `AdaptableUnaryFunction` computes the derivative of a `PolynomialTraits_d::Polynomial_d` with respect to one variable. More...
concept | Evaluate
| This `AdaptableBinaryFunction` evaluates `PolynomialTraits_d::Polynomial_d` with respect to one variable. More...
concept | EvaluateHomogeneous
| This `AdaptableFunctor` provides evaluation of a `PolynomialTraits_d::Polynomial_d` interpreted as a homogeneous polynomial in one variable. More...
concept | GcdUpToConstantFactor
| This `AdaptableBinaryFunction` computes the \( gcd\) up to a constant factor (utcf) of two polynomials of type `PolynomialTraits_d::Polynomial_d`. More...
concept | GetCoefficient
| This `AdaptableBinaryFunction` provides access to coefficients of a `PolynomialTraits_d::Polynomial_d`. More...
concept | GetInnermostCoefficient
| For the given `PolynomialTraits_d::Polynomial_d` this `AdaptableBinaryFunction` returns the coefficient of the (multivariate) monomial specified by the given `CGAL::Exponent_vector`. More...
concept | InnermostLeadingCoefficient
| This `AdaptableUnaryFunction` computes the innermost leading coefficient of a `PolynomialTraits_d::Polynomial_d` \( p\). The innermost leading coefficient is recursively defined as the innermost leading coefficient of the leading coefficient of \( p\). In case \( p\) is univariate it coincides with the leading coefficient. More...
concept | IntegralDivisionUpToConstantFactor
| This `AdaptableBinaryFunction` computes the integral division of two polynomials of type `PolynomialTraits_d::Polynomial_d` up to a constant factor (utcf) . More...
concept | Invert
| This `AdaptableUnaryFunction` inverts one variable in a given `PolynomialTraits_d::Polynomial_d`, that is, for a given polynomial \( p\) it computes \( x^{degree(p)}p(1/x)\). More...
concept | IsSquareFree
| This `AdaptableUnaryFunction` computes whether the given a polynomial of type `PolynomialTraits_d::Polynomial_d` is square free. More...
concept | IsZeroAt
| This `AdaptableFunctor` returns whether a `PolynomialTraits_d::Polynomial_d` \( p\) is zero at a given Cartesian point, which is represented as an iterator range. More...
concept | IsZeroAtHomogeneous
| This `AdaptableFunctor` returns whether a `PolynomialTraits_d::Polynomial_d` \( p\) is zero at a given homogeneous point, which is given by an iterator range. More...
concept | LeadingCoefficient
| This `AdaptableUnaryFunction` computes the leading coefficient of a `PolynomialTraits_d::Polynomial_d`. More...
concept | MakeSquareFree
| This `AdaptableUnaryFunction` computes the square-free part of a polynomial of type `PolynomialTraits_d::Polynomial_d` up to a constant factor. More...
concept | MonomialRepresentation
| This `Functor` outputs the monomial representation of the given polynomial, that is, it writes all non zero terms of the polynomial as `std::pair<CGAL::Exponent_vector, PolynomialTraits_d::Innermost_coefficient_type>` into the given output iterator. More...
concept | Move
| This `AdaptableFunctor` moves a variable at position \( i\) to a new position \( j\). The relative order of the other variables is preserved, that is, the variables between \( x_i\) and \( x_j\) (including \( x_j\)) are moved by one position while \( x_i\) is moved to the former position of \( x_j\). More...
concept | MultivariateContent
| This `AdaptableUnaryFunction` computes the content of a `PolynomialTraits_d::Polynomial_d` with respect to the symmetric view on the polynomial, that is, it computes the gcd of all innermost coefficients. More...
concept | Negate
| This `AdaptableUnaryFunction` computes \( p(-x)\) for a given polynomial \( p\). More...
concept | Permute
| This `AdaptableFunctor` permutes the variables of the given polynomial with respect to a permutation \( \sigma\), that is, each monomial \( \prod x_i^{e_i}\) will be mapped to the monomial \( \prod x_{\sigma(i)}^{e_i}\). The permutation \( \sigma\) is given by the iterator range of length `PolynomialTraits_d::d`, which is supposed to contain the second row of the permutation. More...
concept | PolynomialSubresultants
| Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of type `PolynomialTraits_d::Polynomial_d` with respect to outermost variable. Let \( p=\ccSum{i=0,\ldots,n}{} p_i x^i\) and \( q=\ccSum{i=0,\ldots,m}{} q_i x^i\), where \( x\) is the outermost variable. The \( i\)-th subresultant (with \( i=0,\ldots,\min\{n,m\}\)) is defined by. More...
concept | PolynomialSubresultantsWithCofactors
| Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of degree \( n\) and \( m\), respectively, as defined in the documentation of `PolynomialTraits_d::PolynomialSubresultants`. Moreover, for \( \mathrm{Sres}_i(p,q)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq m-i-1\) and \( \deg v_i\leq n-i-1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i p + v_i q\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Sres}_i(p,q)\). More...
concept | PrincipalSturmHabichtSequence
| Computes the principal leading coefficients of the Sturm-Habicht sequence of a polynomials \( f\) of type `PolynomialTraits_d::Polynomial_d` with respect a certain variable \( x_i\). This means that for the \( j\)-th Sturm-Habicht polynomial, this methods returns the coefficient of \( x_i^j\). More...
concept | PrincipalSubresultants
| Computes the principal subresultant of two polynomials \( p\) and \( q\) of type `PolynomialTraits_d::Coefficient_type` with respect to the outermost variable. The \( i\)-th principal subresultant, \( \mathrm{sres}_i(p,q)\), is defined as the coefficient at \( t^i\) of the \( i\)-th polynomial subresultant \( \mathrm{Sres}_i(p,q)\). Thus, it is either the leading coefficient of \( \mathrm{Sres}_i\), or zero in the case where its degree is below \( i\). More...
concept | PseudoDivision
| This `AdaptableFunctor` computes the pseudo division of two polynomials \( f\) and \( g\). More...
concept | PseudoDivisionQuotient
| This `AdaptableBinaryFunction` computes the quotient of the pseudo division of two polynomials \( f\) and \( g\). More...
concept | PseudoDivisionRemainder
| This `AdaptableBinaryFunction` computes the remainder of the pseudo division of two polynomials \( f\) and \( g\). More...
concept | Resultant
| This `AdaptableBinaryFunction` computes the resultant of two polynomials \( f\) and \( g\) of type `PolynomialTraits_d::Polynomial_d` with respect to a certain variable. More...
concept | Scale
| Given a constant \( c\) this `AdaptableBinaryFunction` scales a `PolynomialTraits_d::Polynomial_d` \( p\) with respect to one variable, that is, it computes \( p(c\cdot x)\). More...
concept | ScaleHomogeneous
| Given a numerator \( a\) and a denominator \( b\) this `AdaptableFunctor` scales a `PolynomialTraits_d::Polynomial_d` \( p\) with respect to one variable, that is, it computes \( b^{degree(p)}\cdot p(a/b\cdot x)\). More...
concept | Shift
| This `AdaptableBinaryFunction` multiplies a `PolynomialTraits_d::Polynomial_d` by the given power of the specified variable. More...
concept | SignAt
| This `AdaptableFunctor` returns the sign of a `PolynomialTraits_d::Polynomial_d` \( p\) at given Cartesian point represented as an iterator range. More...
concept | SignAtHomogeneous
| This `AdaptableFunctor` returns the sign of a `PolynomialTraits_d::Polynomial_d` \( p\) at a given homogeneous point, which is given by an iterator range. More...
concept | SquareFreeFactorize
| This `Functor` computes a square-free factorization of a `PolynomialTraits_d::Polynomial_d`. More...
concept | SquareFreeFactorizeUpToConstantFactor
| This `AdaptableFunctor` computes a square-free factorization up to a constant factor (utcf) of a `PolynomialTraits_d::Polynomial_d`. More...
concept | SturmHabichtSequence
| Computes the Sturm-Habicht sequence (aka the signed subresultant sequence) of a polynomial \( f\) of type `PolynomialTraits_d::Polynomial_d` with respect to a certain variable \( x_i\). The Sturm-Habicht sequence is similar to the polynomial subresultant sequence of \( f\) and its derivative \( f':=\frac{\partial f}{\partial x_i}\) with respect to \( x_i\). The implementation is based on the following definition: More...
concept | SturmHabichtSequenceWithCofactors
| Computes the Sturm-Habicht polynomials of a polynomial \( f\) of degree \( n\), as defined in the documentation of `PolynomialTraits_d::SturmHabichtSequence`. Moreover, for \( \mathrm{Stha}_i(f)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq n-i-2\) and \( \deg v_i\leq n-i-1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i f + v_i f'\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Stha}_i(f)\). More...
concept | Substitute
| This `Functor` substitutes all variables of a given multivariate `PolynomialTraits_d::Polynomial_d` by the values given in the iterator range, where begin refers the value for the innermost variable. More...
concept | SubstituteHomogeneous
| This `Functor` substitutes all variables of a given multivariate `PolynomialTraits_d::Polynomial_d` \( p\) by the values given in the iterator range, where begin refers the value for the innermost variable. In contrast to `PolynomialTraits_d::Substitute` the given polynomial \( p\) is interpreted as a homogeneous polynomial. Hence the iterator range is required to be of length `PolynomialTraits_d::d+1`. More...
concept | Swap
| This `AdaptableFunctor` swaps two variables of a multivariate polynomial. More...
concept | TotalDegree
| This `AdaptableUnaryFunction` computes the total degree of a `PolynomialTraits_d::Polynomial_d`. More...
concept | Translate
| This `AdaptableBinaryFunction` translates a `PolynomialTraits_d::Polynomial_d` with respect to one variable, that is, for a given polynomial \( p\) and constant \( c\) it computes \( p(x+c)\). More...
concept | TranslateHomogeneous
| Given numerator \( a\) and denominator \( b\) this `AdaptableFunctor` translates a `PolynomialTraits_d::Polynomial_d` \( p\) with respect to one variable by \( a/b\), that is, it computes \( b^{degree(p)}\cdot p(x+a/b)\). More...
concept | UnivariateContent
| This `AdaptableUnaryFunction` computes the content of a `PolynomialTraits_d::Polynomial_d` with respect to the univariate (recursive) view on the polynomial, that is, it computes the gcd of all coefficients with respect to one variable. More...
concept | UnivariateContentUpToConstantFactor
| This `AdaptableUnaryFunction` computes the content of a `PolynomialTraits_d::Polynomial_d` with respect to the univariate (recursive) view on the polynomial up to a constant factor (utcf), that is, it computes the \( \mathrm{gcd\_utcf}\) of all coefficients with respect to one variable. More...

## Types
typedef unspecified_type | Polynomial_d
| Type representing \( R[x_0,\dots,x_{d-1}]\).
typedef unspecified_type | Coefficient_type
| Type representing \( R[x_0,\dots,x_{d-2}]\).
typedef unspecified_type | Innermost_coefficient_type
| Type representing the base ring \( R\).
typedef unspecified_type | Coefficient_const_iterator
| Const iterator used to iterate through all coefficients of the polynomial.
typedef unspecified_type | Innermost_coefficient_const_iterator
| Const iterator used to iterate through all innermost coefficients of the polynomial.
template<typename T , int d>
using | Rebind = unspecified_type
| This template class has to define a type `Rebind<T,d>::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`.
typedef unspecified_type | Construct_polynomial
| A model of `PolynomialTraits_d::ConstructPolynomial`.
typedef unspecified_type | Get_coefficient
| A model of `PolynomialTraits_d::GetCoefficient`.
typedef unspecified_type | Get_innermost_coefficient
| A model of `PolynomialTraits_d::GetInnermostCoefficient`.
typedef unspecified_type | Construct_coefficient_const_iterator_range
| A model of `PolynomialTraits_d::ConstructCoefficientConstIteratorRange`.
typedef unspecified_type | Construct_innermost_coefficient_const_iterator_range
| A model of `PolynomialTraits_d::ConstructInnermostCoefficientConstIteratorRange`.
typedef unspecified_type | Swap
| A model of `PolynomialTraits_d::Swap`.
typedef unspecified_type | Move
| A model of `PolynomialTraits_d::Move`.
typedef unspecified_type | Degree
| A model of `PolynomialTraits_d::Degree`.
typedef unspecified_type | Total_degree
| A model of `PolynomialTraits_d::TotalDegree`.
typedef unspecified_type | Degree_vector
| A model of `PolynomialTraits_d::DegreeVector`.
typedef unspecified_type | Leading_coefficient
| A model of `PolynomialTraits_d::LeadingCoefficient`.
typedef unspecified_type | Innermost_leading_coefficient
| A model of `PolynomialTraits_d::InnermostLeadingCoefficient`.
typedef unspecified_type | Canonicalize
| A model of `PolynomialTraits_d::Canonicalize`.
typedef unspecified_type | Differentiate
| A model of `PolynomialTraits_d::Differentiate`.
typedef unspecified_type | Evaluate
| A model of `PolynomialTraits_d::Evaluate`.
typedef unspecified_type | Evaluate_homogeneous
| A model of `PolynomialTraits_d::EvaluateHomogeneous`.
typedef unspecified_type | Substitute
| A model of `PolynomialTraits_d::Substitute`.
typedef unspecified_type | Substitute_homogeneous
| A model of `PolynomialTraits_d::SubstituteHomogeneous`.
typedef unspecified_type | Is_zero_at
| A model of `PolynomialTraits_d::IsZeroAt`.
typedef unspecified_type | Is_zero_at_homogeneous
| A model of `PolynomialTraits_d::IsZeroAtHomogeneous`.
typedef unspecified_type | Sign_at
| A model of `PolynomialTraits_d::SignAt`.
typedef unspecified_type | Sign_at_homogeneous
| A model of `PolynomialTraits_d::SignAtHomogeneous`.
typedef unspecified_type | Compare
| A model of `PolynomialTraits_d::Compare`.
typedef unspecified_type | 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`.
typedef unspecified_type | 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`.
typedef unspecified_type | Shift
| A model of `PolynomialTraits_d::Shift`.
typedef unspecified_type | Negate
| A model of `PolynomialTraits_d::Negate`.
typedef unspecified_type | Invert
| A model of `PolynomialTraits_d::Invert`.
typedef unspecified_type | Translate
| A model of `PolynomialTraits_d::Translate`.
typedef unspecified_type | Translate_homogeneous
| A model of `PolynomialTraits_d::TranslateHomogeneous`.
typedef unspecified_type | Scale
| A model of `PolynomialTraits_d::Scale`.
typedef unspecified_type | Scale_homogeneous
| A model of `PolynomialTraits_d::ScaleHomogeneous`.
typedef unspecified_type | Make_square_free
| A model of `PolynomialTraits_d::MakeSquareFree`.
typedef unspecified_type | 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`.
typedef unspecified_type | Pseudo_division
| A model of `PolynomialTraits_d::PseudoDivision`.
typedef unspecified_type | Pseudo_division_remainder
| A model of `PolynomialTraits_d::PseudoDivisionRemainder`.
typedef unspecified_type | Pseudo_division_quotient
| A model of `PolynomialTraits_d::PseudoDivisionQuotient`.
typedef unspecified_type | Gcd_up_to_constant_factor
| A model of `PolynomialTraits_d::GcdUpToConstantFactor`.
typedef unspecified_type | Integral_division_up_to_constant_factor
| A model of `PolynomialTraits_d::IntegralDivisionUpToConstantFactor`.
typedef unspecified_type | Content_up_to_constant_factor
| A model of `PolynomialTraits_d::UnivariateContentUpToConstantFactor`.
typedef unspecified_type | Square_free_factorize_up_to_constant_factor
| A model of `PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor`.
typedef unspecified_type | Resultant
| A model of `PolynomialTraits_d::Resultant`.
typedef unspecified_type | Polynomial_subresultants
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::PolynomialSubresultants`.
typedef unspecified_type | Polynomial_subresultants_with_cofactors
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::PolynomialSubresultants_with_cofactors`.
typedef unspecified_type | Principal_subresultants
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::PrincipalSubresultants`.
typedef unspecified_type | Sturm_habicht_sequence
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::SturmHabichtSequence`.
typedef unspecified_type | Sturm_habicht_sequence_with_cofactors
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::SturmHabichtSequenceWithCofactors`.
typedef unspecified_type | Principal_sturm_habicht_sequence
| Either `CGAL::Null_functor` or a model of `PolynomialTraits_d::PrincipalSturmHabichtSequence`.

## Constants
static const int | d
| The dimension and the number of variables respectively.

## Member Typedef Documentation

## ◆ Compare

typedef unspecified_type PolynomialTraits_d::Compare

A model of `PolynomialTraits_d::Compare`.

In case `Innermost_coefficient_type` is not `RealEmbeddable` this is `CGAL::Null_functor`.

## ◆ Rebind

template<typename T , int d>

using PolynomialTraits_d::Rebind = unspecified_type

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

Note
  It can be implemented using a nested template class.

## ◆ Sign_at

typedef unspecified_type PolynomialTraits_d::Sign_at

A model of `PolynomialTraits_d::SignAt`.

In case `Innermost_coefficient_type` is not `RealEmbeddable` this is `CGAL::Null_functor`.

## ◆ Sign_at_homogeneous

typedef unspecified_type PolynomialTraits_d::Sign_at_homogeneous

A model of `PolynomialTraits_d::SignAtHomogeneous`.

In case `Innermost_coefficient_type` is not `RealEmbeddable` this is `CGAL::Null_functor`.