\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.11.2 - Polynomial
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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

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>.

Adapts Polynomial_traits_d::Canonicalize.

For more details see the concept PolynomialTraits_d::Canonicalize.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Canonicalize

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/gcd_up_to_constant_factor.cpp.
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>.

Adapts Polynomial_traits_d::Compare.

For more details see the concept PolynomialTraits_d::Compare.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Compare

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Degree.

For more details see the concept PolynomialTraits_d::Degree.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Degree

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/degree.cpp.
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>.

Adapts Polynomial_traits_d::DegreeVector.

For more details see the concept PolynomialTraits_d::DegreeVector.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Degree_vector

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/degree.cpp.
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>.

Adapts Polynomial_traits_d::Differentiate.

For more details see the concept PolynomialTraits_d::Differentiate.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Differentiate

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Evaluate.

For more details see the concept PolynomialTraits_d::Evaluate.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Evaluate

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/substitute.cpp.
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>.

Adapts Polynomial_traits_d::Evaluate_homogeneous.

For more details see the concept PolynomialTraits_d::EvaluateHomogeneous.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::EvaluateHomogeneous

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/coefficient_access.cpp.
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>.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::InnermostLeadingCoefficient.

For more details see the concept PolynomialTraits_d::InnermostLeadingCoefficient.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::InnermostLeadingCoefficient

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Invert.

For more details see the concept PolynomialTraits_d::Invert.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Invert

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Leading_coefficient.

For more details see the concept PolynomialTraits_d::LeadingCoefficient.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::LeadingCoefficient

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Move.

For more details see the concept PolynomialTraits_d::Move.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Move

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/swap_move.cpp.
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>.

Adapts Polynomial_traits_d::Multivariate_content.

For more details see the concept PolynomialTraits_d::MultivariateContent.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::MultivariateContent

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Negate.

For more details see the concept PolynomialTraits_d::Negate.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Negate

#include <CGAL/polynomial_utils.h>

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

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

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/subresultants.cpp.
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.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Permute.

For more details see the concept PolynomialTraits_d::Permute.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Permute

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/swap_move.cpp.
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.

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

#include <CGAL/polynomial_utils.h>

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.

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

#include <CGAL/polynomial_utils.h>

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.

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

#include <CGAL/polynomial_utils.h>

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.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Pseudo_division.

For more details see the concept PolynomialTraits_d::PseudoDivision.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::PseudoDivision

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Resultant.

For more details see the concept PolynomialTraits_d::Resultant.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Resultant

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/subresultants.cpp.
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>.

Adapts Polynomial_traits_d::Scale.

For more details see the concept PolynomialTraits_d::Scale.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Scale

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Scale_homogeneous.

For more details see the concept PolynomialTraits_d::ScaleHomogeneous.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::ScaleHomogeneous

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Shift.

For more details see the concept PolynomialTraits_d::Shift.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Shift

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/construction.cpp, and Polynomial/gcd_up_to_constant_factor.cpp.
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>.

Adapts Polynomial_traits_d::Sign_at.

For more details see the concept PolynomialTraits_d::SignAt.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SignAt

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>

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.

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

#include <CGAL/polynomial_utils.h>

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.

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

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Substitute.

For more details see the concept PolynomialTraits_d::Substitute.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Substitute

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/substitute.cpp.
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>.

Adapts Polynomial_traits_d::Substitute_homogeneous.

For more details see the concept PolynomialTraits_d::SubstituteHomogeneous.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::SubstituteHomogeneous

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Swap.

For more details see the concept PolynomialTraits_d::Swap.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Swap

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/swap_move.cpp.
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>.

Adapts Polynomial_traits_d::Total_degree.

For more details see the concept PolynomialTraits_d::TotalDegree.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::TotalDegree

#include <CGAL/polynomial_utils.h>

Examples:
Polynomial/degree.cpp.
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>.

Adapts Polynomial_traits_d::Translate.

For more details see the concept PolynomialTraits_d::Translate.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Translate

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::Translate_homogeneous.

For more details see the concept PolynomialTraits_d::TranslateHomogeneous.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::TranslateHomogeneous

#include <CGAL/polynomial_utils.h>

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>.

Adapts Polynomial_traits_d::UnivariateContent.

For more details see the concept PolynomialTraits_d::UnivariateContent.

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::Univariate_Content

#include <CGAL/polynomial_utils.h>

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>.

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

#include <CGAL/polynomial_utils.h>