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CGAL 4.14.2 - Polynomial

Functions

template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Canonicalize::result_type CGAL::canonicalize (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Compare::result_type CGAL::compare (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Degree::result_type CGAL::degree (const Polynomial_d &p, int i, index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Degree_vector::result_type CGAL::degree_vector (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Differentiate::result_type CGAL::differentiate (const Polynomial_d &p, index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Evaluate_homogeneous::result_type CGAL::evaluate_homogeneous (const Polynomial_d &p, Polynomial_traits_d< Polynomial_d >::Coefficient_type u, Polynomial_traits_d< Polynomial_d >::Coefficient_type v)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Evaluate::result_type CGAL::evaluate (const Polynomial_d &p, Polynomial_traits_d< Polynomial_d >::Coefficient_type x)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Gcd_up_to_constant_factor::result_type CGAL::gcd_up_to_constant_factor (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::get_coefficient::result_type CGAL::get_coefficient (const Polynomial_d &p, int i)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::get_innermost_coefficient::result_type CGAL::get_innermost_coefficient (const Polynomial_d &p, Exponent_vector ev)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Innermost_leading_coefficient::result_type CGAL::innermost_leading_coefficient (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Integral_division_up_to_constant_factor::result_type CGAL::integral_division_up_to_constant_factor (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Invert::result_type CGAL::invert (const Polynomial_d &p, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Is_square_free::result_type CGAL::is_square_free (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class InputIterator >
Polynomial_traits_d< Polynomial_d >::Is_zero_at_homogeneous::result_type CGAL::is_zero_at_homogeneous (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class InputIterator >
Polynomial_traits_d< Polynomial_d >::Is_zero_at::result_type CGAL::is_zero_at (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Leading_coefficient::result_type CGAL::leading_coefficient (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Make_square_free::result_type CGAL::make_square_free (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Move::result_type CGAL::move (const Polynomial_d &p, int i, int j)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Multivariate_content::result_type CGAL::multivariate_content (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Negate::result_type CGAL::negate (const Polynomial_d &p, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<typename Polynomial_d >
int CGAL::number_of_real_roots (Polynomial_d f)
 computes the number of distinct real roots of \(f\). More...
 
template<typename InputIterator >
int CGAL::number_of_real_roots (InputIterator start, InputIterator end)
 computes the number of distinct real roots of \( f\) whose principal Sturm-Habicht coefficients are passed by the iterator range. More...
 
template<class Polynomial_d , class InputIterator >
Polynomial_traits_d< Polynomial_d >::Permute::result_type CGAL::permute (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::polynomial_subresultants (Polynomial_d p, Polynomial_d q, OutputIterator out)
 computes the polynomial subresultants of \( p\) and \( q\), with respect to the outermost variable. More...
 
template<typename Polynomial_d , typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 CGAL::polynomial_subresultants_with_cofactors (Polynomial_d p, Polynomial_d q, OutputIterator1 sres_out, OutputIterator2 coP_out, OutputIterator3 coQ_out)
 computes the polynomial subresultants of \( p\) and \( q\), sres_out, with respect to the outermost variable, and the cofactors for \( P\), coP_out and \( Q\), coQ_out. More...
 
template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::principal_sturm_habicht_sequence (typename Polynomial_d f, OutputIterator out)
 computes the principal Sturm-Habicht coefficients of \( f\) with respect to the outermost variable. More...
 
template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::principal_subresultants (Polynomial_d p, Polynomial_d q, OutputIterator out)
 computes the principal subresultants of \( p\) and \( q\), with respect to the outermost variable. More...
 
template<class Polynomial_d >
void CGAL::pseudo_division (const Polynomial_d &f, const Polynomial_d &g, Polynomial_d &q, Polynomial_d &r, Polynomial_traits_d< Polynomial_d >::Coefficient_type &D)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Pseudo_division_quotient::result_type CGAL::pseudo_division_quotient (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Pseudo_division_remainder::result_type CGAL::pseudo_division_remainder (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Resultant::result_type CGAL::resultant (const Polynomial_d &p, const Polynomial_d &q)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Scale_homogeneous::result_type CGAL::scale_homogeneous (const Polynomial_d &p, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &u, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &v, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Scale::result_type CGAL::scale (const Polynomial_d &p, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &a, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Shift::result_type CGAL::shift (const Polynomial_d &p, int i, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class InputIterator >
Polynomial_traits_d< Polynomial_d >::Sign_at_homogeneous::result_type CGAL::sign_at_homogeneous (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class InputIterator >
Polynomial_traits_d< Polynomial_d >::Sign_at::result_type CGAL::sign_at (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize (const Polynomial_d &p, OutputIterator it, Polynomial_traits_d< Polynomial >::Innermost_coefficient &a)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize (const Polynomial_d &p, OutputIterator it)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize_up_to_constant_factor (const Polynomial_d &p, OutputIterator it)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::sturm_habicht_sequence (Polynomial_d f, OutputIterator out)
 computes the Sturm-Habicht-sequence of \( f\) with respect to the outermost variable. More...
 
template<typename Polynomial_d , typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 CGAL::sturm_habicht_sequence_with_cofactors (Polynomial_d f, OutputIterator1 stha_out, OutputIterator2 cof_out, OutputIterator3 cofx_out)
 computes the Sturm-Habicht sequence of \( f\) stha_out, with respect to the outermost variable, and the cofactors for \( f\), cof_out and \( f'\), cofx_out. More...
 
template<class Polynomial_d , class InputIterator >
CGAL::Coercion_traits< Polynomial_traits_d< Polynomial_d >::Innermost_coefficient, std::iterator_traits< Input_iterator >::value_type >::Type CGAL::substitute_homogeneous (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d , class InputIterator >
CGAL::Coercion_traits< Polynomial_traits_d< Polynomial_d >::Innermost_coefficient, std::iterator_traits< Input_iterator >::value_type >::Type CGAL::substitute (const Polynomial_d &p, InputIterator begin, InputIterator end)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Swap::result_type CGAL::swap (const Polynomial_d &p, int i, int j)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Total_degree::result_type CGAL::total_degree (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Translate_homogeneous::result_type CGAL::translate_homogeneous (const Polynomial_d &p, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &u, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &v, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Translate::result_type CGAL::translate (const Polynomial_d &p, const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &a, int index=Polynomial_traits_d< Polynomial_d >::d-1)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Univariate_content::result_type CGAL::univariate_content (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 
template<class Polynomial_d >
Polynomial_traits_d< Polynomial_d >::Univariate_content_up_to_constant_factor::result_type CGAL::univariate_content_up_to_constant_factor (const Polynomial_d &p)
 For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>. More...
 

Function Documentation

◆ canonicalize()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Canonicalize::result_type CGAL::canonicalize ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Canonicalize.

For more details see the concept PolynomialTraits_d::Canonicalize.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Canonicalize
Examples:
Polynomial/gcd_up_to_constant_factor.cpp.

◆ compare()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Compare::result_type CGAL::compare ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Compare.

For more details see the concept PolynomialTraits_d::Compare.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Compare

◆ degree()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Degree::result_type CGAL::degree ( const Polynomial_d p,
int  i,
index  = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Degree.

For more details see the concept PolynomialTraits_d::Degree.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Degree
Examples:
Polynomial/degree.cpp.

◆ degree_vector()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Degree_vector::result_type CGAL::degree_vector ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::DegreeVector.

For more details see the concept PolynomialTraits_d::DegreeVector.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Degree_vector
Examples:
Polynomial/degree.cpp.

◆ differentiate()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Differentiate::result_type CGAL::differentiate ( const Polynomial_d p,
index  = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Differentiate.

For more details see the concept PolynomialTraits_d::Differentiate.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Differentiate

◆ evaluate()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Evaluate::result_type CGAL::evaluate ( const Polynomial_d p,
Polynomial_traits_d< Polynomial_d >::Coefficient_type  x 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Evaluate.

For more details see the concept PolynomialTraits_d::Evaluate.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Evaluate
Examples:
Polynomial/substitute.cpp.

◆ evaluate_homogeneous()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Evaluate_homogeneous::result_type CGAL::evaluate_homogeneous ( const Polynomial_d p,
Polynomial_traits_d< Polynomial_d >::Coefficient_type  u,
Polynomial_traits_d< Polynomial_d >::Coefficient_type  v 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Evaluate_homogeneous.

For more details see the concept PolynomialTraits_d::EvaluateHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::EvaluateHomogeneous

◆ gcd_up_to_constant_factor()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Gcd_up_to_constant_factor::result_type CGAL::gcd_up_to_constant_factor ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Gcd_up_to_constant_factor.

For more details see the concept PolynomialTraits_d::GcdUpToConstantFactor.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::GcdUpToConstantFactor

◆ get_coefficient()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::get_coefficient::result_type CGAL::get_coefficient ( const Polynomial_d p,
int  i 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::GetCoefficient.

For more details see the concept PolynomialTraits_d::GetCoefficient.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::GetCoefficient
PolynomialTraits_d::GetInnermostCoefficient
Examples:
Polynomial/coefficient_access.cpp.

◆ get_innermost_coefficient()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::get_innermost_coefficient::result_type CGAL::get_innermost_coefficient ( const Polynomial_d p,
Exponent_vector  ev 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::GetInnermostCoefficient.

For more details see the concept PolynomialTraits_d::GetInnermostCoefficient.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::GetCoefficient
PolynomialTraits_d::GetInnermostCoefficient

◆ innermost_leading_coefficient()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Innermost_leading_coefficient::result_type CGAL::innermost_leading_coefficient ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::InnermostLeadingCoefficient.

For more details see the concept PolynomialTraits_d::InnermostLeadingCoefficient.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::InnermostLeadingCoefficient

◆ integral_division_up_to_constant_factor()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Integral_division_up_to_constant_factor::result_type CGAL::integral_division_up_to_constant_factor ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Integral_division_up_to_constant_factor.

For more details see the concept PolynomialTraits_d::IntegralDivisionUpToConstantFactor.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::IntegralDivisionUpToConstantFactor

◆ invert()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Invert::result_type CGAL::invert ( const Polynomial_d p,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Invert.

For more details see the concept PolynomialTraits_d::Invert.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Invert

◆ is_square_free()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Is_square_free::result_type CGAL::is_square_free ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Is_square_free.

For more details see the concept PolynomialTraits_d::IsSquareFree.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::IsSquareFree

◆ is_zero_at()

template<class Polynomial_d , class InputIterator >
Polynomial_traits_d<Polynomial_d>::Is_zero_at::result_type CGAL::is_zero_at ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Is_zero_at.

For more details see the concept PolynomialTraits_d::IsZeroAt.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::IsZeroAt

◆ is_zero_at_homogeneous()

template<class Polynomial_d , class InputIterator >
Polynomial_traits_d<Polynomial_d>::Is_zero_at_homogeneous::result_type CGAL::is_zero_at_homogeneous ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Is_zero_at_homogeneous.

For more details see the concept PolynomialTraits_d::IsZeroAtHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::IsZeroAtHomogeneous

◆ leading_coefficient()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Leading_coefficient::result_type CGAL::leading_coefficient ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Leading_coefficient.

For more details see the concept PolynomialTraits_d::LeadingCoefficient.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::LeadingCoefficient

◆ make_square_free()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Make_square_free::result_type CGAL::make_square_free ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Make_square_free.

For more details see the concept PolynomialTraits_d::MakeSquareFree.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::MakeSquareFree

◆ move()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Move::result_type CGAL::move ( const Polynomial_d p,
int  i,
int  j 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Move.

For more details see the concept PolynomialTraits_d::Move.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Move
Examples:
Polynomial/swap_move.cpp.

◆ multivariate_content()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Multivariate_content::result_type CGAL::multivariate_content ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Multivariate_content.

For more details see the concept PolynomialTraits_d::MultivariateContent.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::MultivariateContent

◆ negate()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Negate::result_type CGAL::negate ( const Polynomial_d p,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Negate.

For more details see the concept PolynomialTraits_d::Negate.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Negate

◆ number_of_real_roots() [1/2]

template<typename Polynomial_d >
int CGAL::number_of_real_roots ( Polynomial_d  f)

#include <CGAL/polynomial_utils.h>

computes the number of distinct real roots of \(f\).

Given a polynomial \( f\), or a range of values that is interpreted as the principal Sturm-Habicht coefficients of \( f\), the function computes

\[ m:=\# \{\alpha\in\mathbb{R}\mid f(\alpha)=0\} \]

that is, the number of distinct real roots of \( f\).

The coefficient type of the polynomial, or the value type of the iterator range, respectively must be a model of RealEmbeddable. In the second version, it is not required to pass the exact princiapl Sturm-Habicht coefficients to the functions; it is only required that the sign of each element corresponds to the sign of the actual principal Sturm-Habicht coefficient.

Advanced

We explain the internals of this function. For a sequence \( I:=(a_0,\ldots,a_n)\) of real numbers with \( a_0\neq 0\), define

\[ C(I)=\ccSum{i=1}{s}\epsilon_i \]

where \( s\) is the number of subsequences of \( I\) of the form

underbrace.png

with \( a\neq 0,b\neq 0, k\geq 0\).

For the \( i\)-th subsequence of \( I\), define

\[ \epsilon_i:=\begin{array}{cc} 0 & \mbox{if $k$ is odd},\\ (-1)^{k/2}\mathrm{sign}(ab) & \mbox{if $k$ is even}. \end{array} \]

For \( f\in\mathbb{R}[x]\) with \( \deg f=n\), we have:

\[ C(\mathrm{stha}_n(f),\ldots,\mathrm{stha}_0(f)) = \#\{\alpha\in\R\mid f(\alpha)=0\} \]

In other words, the signs of the principal Sturm-Habicht coefficients determine the number of distinct real roots of \( f\).

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PrincipalSturmHabichtSequence
Examples:
Polynomial/subresultants.cpp.

◆ number_of_real_roots() [2/2]

template<typename InputIterator >
int CGAL::number_of_real_roots ( InputIterator  start,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

computes the number of distinct real roots of \( f\) whose principal Sturm-Habicht coefficients are passed by the iterator range.

Given a polynomial \( f\), or a range of values that is interpreted as the principal Sturm-Habicht coefficients of \( f\), the function computes

\[ m:=\# \{\alpha\in\mathbb{R}\mid f(\alpha)=0\} \]

that is, the number of distinct real roots of \( f\).

The coefficient type of the polynomial, or the value type of the iterator range, respectively must be a model of RealEmbeddable. In the second version, it is not required to pass the exact princiapl Sturm-Habicht coefficients to the functions; it is only required that the sign of each element corresponds to the sign of the actual principal Sturm-Habicht coefficient.

Advanced

We explain the internals of this function. For a sequence \( I:=(a_0,\ldots,a_n)\) of real numbers with \( a_0\neq 0\), define

\[ C(I)=\ccSum{i=1}{s}\epsilon_i \]

where \( s\) is the number of subsequences of \( I\) of the form

underbrace.png

with \( a\neq 0,b\neq 0, k\geq 0\).

For the \( i\)-th subsequence of \( I\), define

\[ \epsilon_i:=\begin{array}{cc} 0 & \mbox{if $k$ is odd},\\ (-1)^{k/2}\mathrm{sign}(ab) & \mbox{if $k$ is even}. \end{array} \]

For \( f\in\mathbb{R}[x]\) with \( \deg f=n\), we have:

\[ C(\mathrm{stha}_n(f),\ldots,\mathrm{stha}_0(f)) = \#\{\alpha\in\R\mid f(\alpha)=0\} \]

In other words, the signs of the principal Sturm-Habicht coefficients determine the number of distinct real roots of \( f\).

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PrincipalSturmHabichtSequence

◆ permute()

template<class Polynomial_d , class InputIterator >
Polynomial_traits_d<Polynomial_d>::Permute::result_type CGAL::permute ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Permute.

For more details see the concept PolynomialTraits_d::Permute.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Permute
Examples:
Polynomial/swap_move.cpp.

◆ polynomial_subresultants()

template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::polynomial_subresultants ( Polynomial_d  p,
Polynomial_d  q,
OutputIterator  out 
)

#include <CGAL/polynomial_utils.h>

computes the polynomial subresultants of \( p\) and \( q\), with respect to the outermost variable.

Each element is of type Polynomial_d.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::PolynomialSubresultants.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PolynomialSubresultants

◆ polynomial_subresultants_with_cofactors()

template<typename Polynomial_d , typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 CGAL::polynomial_subresultants_with_cofactors ( Polynomial_d  p,
Polynomial_d  q,
OutputIterator1  sres_out,
OutputIterator2  coP_out,
OutputIterator3  coQ_out 
)

#include <CGAL/polynomial_utils.h>

computes the polynomial subresultants of \( p\) and \( q\), sres_out, with respect to the outermost variable, and the cofactors for \( P\), coP_out and \( Q\), coQ_out.

The elements of each output range are of type Polynomial_d.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::PolynomialSubresultantsWithCofactors.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PolynomialSubresultantsWithCofactors

◆ principal_sturm_habicht_sequence()

template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::principal_sturm_habicht_sequence ( typename Polynomial_d  f,
OutputIterator  out 
)

#include <CGAL/polynomial_utils.h>

computes the principal Sturm-Habicht coefficients of \( f\) with respect to the outermost variable.

Each element is of type Polynomial_traits_d::Coefficient_typeb.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::PrincipalSturmHabichtSequence.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PrincipalSturmHabichtSequence

◆ principal_subresultants()

template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::principal_subresultants ( Polynomial_d  p,
Polynomial_d  q,
OutputIterator  out 
)

#include <CGAL/polynomial_utils.h>

computes the principal subresultants of \( p\) and \( q\), with respect to the outermost variable.

Each element is of type Polynomial_traits_d::Coefficient_type.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::PrincipalSubresultants.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PrincipalSubresultants

◆ pseudo_division()

template<class Polynomial_d >
void CGAL::pseudo_division ( const Polynomial_d f,
const Polynomial_d g,
Polynomial_d q,
Polynomial_d r,
Polynomial_traits_d< Polynomial_d >::Coefficient_type &  D 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Pseudo_division.

For more details see the concept PolynomialTraits_d::PseudoDivision.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PseudoDivision

◆ pseudo_division_quotient()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Pseudo_division_quotient::result_type CGAL::pseudo_division_quotient ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Pseudo_division_quotient.

For more details see the concept PolynomialTraits_d::PseudoDivisionQuotient.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PseudoDivisionQuotient

◆ pseudo_division_remainder()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Pseudo_division_remainder::result_type CGAL::pseudo_division_remainder ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Pseudo_division_remainder.

For more details see the concept PolynomialTraits_d::PseudoDivisionRemainder.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PseudoDivisionRemainder

◆ resultant()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Resultant::result_type CGAL::resultant ( const Polynomial_d p,
const Polynomial_d q 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Resultant.

For more details see the concept PolynomialTraits_d::Resultant.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Resultant
Examples:
Polynomial/subresultants.cpp.

◆ scale()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Scale::result_type CGAL::scale ( const Polynomial_d p,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  a,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Scale.

For more details see the concept PolynomialTraits_d::Scale.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Scale

◆ scale_homogeneous()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Scale_homogeneous::result_type CGAL::scale_homogeneous ( const Polynomial_d p,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  u,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  v,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Scale_homogeneous.

For more details see the concept PolynomialTraits_d::ScaleHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::ScaleHomogeneous

◆ shift()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Shift::result_type CGAL::shift ( const Polynomial_d p,
int  i,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Shift.

For more details see the concept PolynomialTraits_d::Shift.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Shift
Examples:
Polynomial/construction.cpp, and Polynomial/gcd_up_to_constant_factor.cpp.

◆ sign_at()

template<class Polynomial_d , class InputIterator >
Polynomial_traits_d<Polynomial_d>::Sign_at::result_type CGAL::sign_at ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Sign_at.

For more details see the concept PolynomialTraits_d::SignAt.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SignAt

◆ sign_at_homogeneous()

template<class Polynomial_d , class InputIterator >
Polynomial_traits_d<Polynomial_d>::Sign_at_homogeneous::result_type CGAL::sign_at_homogeneous ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Sign_at_homogeneous.

For more details see the concept PolynomialTraits_d::SignAtHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SignAtHomogeneous

◆ square_free_factorize() [1/2]

template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize ( const Polynomial_d p,
OutputIterator  it,
Polynomial_traits_d< Polynomial >::Innermost_coefficient &  a 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Square_free_factorize.

For more details see the concept PolynomialTraits_d::SquareFreeFactorize.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SquareFreeFactorize

◆ square_free_factorize() [2/2]

template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize ( const Polynomial_d p,
OutputIterator  it 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Square_free_factorize.

For more details see the concept PolynomialTraits_d::SquareFreeFactorize.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SquareFreeFactorize

◆ square_free_factorize_up_to_constant_factor()

template<class Polynomial_d , class OutputIterator >
OutputIterator CGAL::square_free_factorize_up_to_constant_factor ( const Polynomial_d p,
OutputIterator  it 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Square_free_factorize_up_to_constant_factor.

For more details see the concept PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor

◆ sturm_habicht_sequence()

template<typename Polynomial_d , typename OutputIterator >
OutputIterator CGAL::sturm_habicht_sequence ( Polynomial_d  f,
OutputIterator  out 
)

#include <CGAL/polynomial_utils.h>

computes the Sturm-Habicht-sequence of \( f\) with respect to the outermost variable.

Each element is of type Polynomial_d.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::SturmHabichtSequence.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SturmHabichtSequence

◆ sturm_habicht_sequence_with_cofactors()

template<typename Polynomial_d , typename OutputIterator1 , typename OutputIterator2 , typename OutputIterator3 >
OutputIterator1 CGAL::sturm_habicht_sequence_with_cofactors ( Polynomial_d  f,
OutputIterator1  stha_out,
OutputIterator2  cof_out,
OutputIterator3  cofx_out 
)

#include <CGAL/polynomial_utils.h>

computes the Sturm-Habicht sequence of \( f\) stha_out, with respect to the outermost variable, and the cofactors for \( f\), cof_out and \( f'\), cofx_out.

The elements of each output range are of type Polynomial_d.

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

For more details see the concept PolynomialTraits_d::SturmHabichtSequenceWithCofactors.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SturmHabichtSequenceWithCofactors

◆ substitute()

template<class Polynomial_d , class InputIterator >
CGAL::Coercion_traits<Polynomial_traits_d<Polynomial_d>::Innermost_coefficient,std::iterator_traits<Input_iterator>::value_type>::Type CGAL::substitute ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Substitute.

For more details see the concept PolynomialTraits_d::Substitute.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Substitute
Examples:
Polynomial/substitute.cpp.

◆ substitute_homogeneous()

template<class Polynomial_d , class InputIterator >
CGAL::Coercion_traits<Polynomial_traits_d<Polynomial_d>::Innermost_coefficient,std::iterator_traits<Input_iterator>::value_type>::Type CGAL::substitute_homogeneous ( const Polynomial_d p,
InputIterator  begin,
InputIterator  end 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Substitute_homogeneous.

For more details see the concept PolynomialTraits_d::SubstituteHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SubstituteHomogeneous

◆ swap()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Swap::result_type CGAL::swap ( const Polynomial_d p,
int  i,
int  j 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Swap.

For more details see the concept PolynomialTraits_d::Swap.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Swap
Examples:
Polynomial/swap_move.cpp.

◆ total_degree()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Total_degree::result_type CGAL::total_degree ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Total_degree.

For more details see the concept PolynomialTraits_d::TotalDegree.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::TotalDegree
Examples:
Polynomial/degree.cpp.

◆ translate()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Translate::result_type CGAL::translate ( const Polynomial_d p,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  a,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Translate.

For more details see the concept PolynomialTraits_d::Translate.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Translate

◆ translate_homogeneous()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Translate_homogeneous::result_type CGAL::translate_homogeneous ( const Polynomial_d p,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  u,
const Polynomial_traits_d< Polynomial_d >::Innermost_coefficient_type &  v,
int  index = Polynomial_traits_dPolynomial_d >::d-1 
)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Translate_homogeneous.

For more details see the concept PolynomialTraits_d::TranslateHomogeneous.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::TranslateHomogeneous

◆ univariate_content()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Univariate_content::result_type CGAL::univariate_content ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::UnivariateContent.

For more details see the concept PolynomialTraits_d::UnivariateContent.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Univariate_Content

◆ univariate_content_up_to_constant_factor()

template<class Polynomial_d >
Polynomial_traits_d<Polynomial_d>::Univariate_content_up_to_constant_factor::result_type CGAL::univariate_content_up_to_constant_factor ( const Polynomial_d p)

#include <CGAL/polynomial_utils.h>

For a given Polynomial_d, adapts the according functor in Polynomial_traits_d<Polynomial_d>.

Adapts Polynomial_traits_d::Univariate_content_up_to_constant_factor.

For more details see the concept PolynomialTraits_d::UnivariateContentUpToConstantFactor.

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::UnivariateContentUpToConstantFactor