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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

template<typename Polynomial_d >
int	CGAL::number_of_real_roots (Polynomial_d f)
	computes the number of distinct real roots of $f$.

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.

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>`.

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.

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`.

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.

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.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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.

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`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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>`.

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

◆ 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_d< Polynomial_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

◆ 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

◆ differentiate()

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`
	)

#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

◆ 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

◆ 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_d< Polynomial_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

◆ 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_d< Polynomial_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 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

\[ (a,\underbrace{0, \dots,0}_{k},b) \]

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

◆ number_of_real_roots() [2/2]

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

\[ (a,\underbrace{0, \dots,0}_{k},b) \]

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.

◆ 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

◆ 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

◆ 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_d< Polynomial_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_d< Polynomial_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_d< Polynomial_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

◆ 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
	)

#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,
		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_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

◆ 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

◆ 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

◆ 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_d< Polynomial_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_d< Polynomial_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

Functions

Function Documentation

◆ canonicalize()

◆ compare()

◆ degree()

◆ degree_vector()

◆ differentiate()

◆ evaluate()

◆ evaluate_homogeneous()

◆ gcd_up_to_constant_factor()

◆ get_coefficient()

◆ get_innermost_coefficient()

◆ innermost_leading_coefficient()

◆ integral_division_up_to_constant_factor()

◆ invert()

◆ is_square_free()

◆ is_zero_at()

◆ is_zero_at_homogeneous()

◆ leading_coefficient()

◆ make_square_free()

◆ move()

◆ multivariate_content()

◆ negate()

◆ number_of_real_roots() [1/2]

◆ number_of_real_roots() [2/2]

◆ permute()

◆ polynomial_subresultants()

◆ polynomial_subresultants_with_cofactors()

◆ principal_sturm_habicht_sequence()

◆ principal_subresultants()

◆ pseudo_division()

◆ pseudo_division_quotient()

◆ pseudo_division_remainder()

◆ resultant()

◆ scale()

◆ scale_homogeneous()

◆ shift()

◆ sign_at()

◆ sign_at_homogeneous()

◆ square_free_factorize() [1/2]

◆ square_free_factorize() [2/2]

◆ square_free_factorize_up_to_constant_factor()

◆ sturm_habicht_sequence()

◆ sturm_habicht_sequence_with_cofactors()

◆ substitute()

◆ substitute_homogeneous()

◆ swap()

◆ total_degree()

◆ translate()

◆ translate_homogeneous()

◆ univariate_content()

◆ univariate_content_up_to_constant_factor()