CGAL 5.1.3  Polynomial

Concepts  
concept  Polynomial_d 
A model of Polynomial_d is representing a multivariate polynomial in \( d \geq 1\) variables over some basic ring \( R\). This type is denoted as the innermost coefficient. A model of Polynomial_d must be accompanied by a traits class CGAL::Polynomial_traits_d<Polynomial_d> , which is a model of PolynomialTraits_d . Please have a look at the concept PolynomialTraits_d , since nearly all functionality related to polynomials is provided by the traits. More...  
concept  PolynomialTraits_d::Canonicalize 
For a given polynomial \( p\) this AdaptableUnaryFunction computes the unique representative of the set \[ {\cal P} := \{ q\ \ \lambda * q = p\ for\ some\ \lambda \in R \}, \] where \( R\) is the base of the polynomial ring. More...  
concept  PolynomialTraits_d::Compare 
This AdaptableBinaryFunction compares two polynomials, with respect to the lexicographic order with preference to the outermost variable. More...  
concept  PolynomialTraits_d::ConstructCoefficientConstIteratorRange 
This AdaptableUnaryFunction returns a const iterator range over the coefficients of the given polynomial, with respect to the outermost variable, \( x_{d1}\). The range starts with the coefficient for \( x_{d1}^0\). More...  
concept  PolynomialTraits_d::ConstructInnermostCoefficientConstIteratorRange 
This AdaptableUnaryFunction returns a const iterator range over all innermost coefficients of the given polynomial. More...  
concept  PolynomialTraits_d::ConstructPolynomial 
This AdaptableFunctor provides several operators to construct objects of type PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::Degree 
This AdaptableUnaryFunction computes the degree of a PolynomialTraits_d::Polynomial_d with respect to a certain variable. More...  
concept  PolynomialTraits_d::DegreeVector 
For a given PolynomialTraits_d::Polynomial_d \( p\) this AdaptableUnaryFunction returns the degree vector, that is, it returns the exponent vector of the monomial of highest order in \( p\), where the monomial order is the lexicographic order giving outer variables a higher priority. In particular, this is the monomial that belongs to the innermost leading coefficient of \( p\). More...  
concept  PolynomialTraits_d::Differentiate 
This AdaptableUnaryFunction computes the derivative of a PolynomialTraits_d::Polynomial_d with respect to one variable. More...  
concept  PolynomialTraits_d::Evaluate 
This AdaptableBinaryFunction evaluates PolynomialTraits_d::Polynomial_d with respect to one variable. More...  
concept  PolynomialTraits_d::EvaluateHomogeneous 
This AdaptableFunctor provides evaluation of a PolynomialTraits_d::Polynomial_d interpreted as a homogeneous polynomial in one variable. More...  
concept  PolynomialTraits_d::GcdUpToConstantFactor 
This AdaptableBinaryFunction computes the \( gcd\) up to a constant factor (utcf) of two polynomials of type PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::GetCoefficient 
This AdaptableBinaryFunction provides access to coefficients of a PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::GetInnermostCoefficient 
For the given PolynomialTraits_d::Polynomial_d this AdaptableBinaryFunction returns the coefficient of the (multivariate) monomial specified by the given CGAL::Exponent_vector . More...  
concept  PolynomialTraits_d::InnermostLeadingCoefficient 
This AdaptableUnaryFunction computes the innermost leading coefficient of a PolynomialTraits_d::Polynomial_d \( p\). The innermost leading coefficient is recursively defined as the innermost leading coefficient of the leading coefficient of \( p\). In case \( p\) is univariate it coincides with the leading coefficient. More...  
concept  PolynomialTraits_d::IntegralDivisionUpToConstantFactor 
This AdaptableBinaryFunction computes the integral division of two polynomials of type PolynomialTraits_d::Polynomial_d up to a constant factor (utcf) . More...  
concept  PolynomialTraits_d::Invert 
This AdaptableUnaryFunction inverts one variable in a given PolynomialTraits_d::Polynomial_d , that is, for a given polynomial \( p\) it computes \( x^{degree(p)}p(1/x)\). More...  
concept  PolynomialTraits_d::IsSquareFree 
This AdaptableUnaryFunction computes whether the given a polynomial of type PolynomialTraits_d::Polynomial_d is square free. More...  
concept  PolynomialTraits_d::IsZeroAt 
This AdaptableFunctor returns whether a PolynomialTraits_d::Polynomial_d \( p\) is zero at a given Cartesian point, which is represented as an iterator range. More...  
concept  PolynomialTraits_d::IsZeroAtHomogeneous 
This AdaptableFunctor returns whether a PolynomialTraits_d::Polynomial_d \( p\) is zero at a given homogeneous point, which is given by an iterator range. More...  
concept  PolynomialTraits_d::LeadingCoefficient 
This AdaptableUnaryFunction computes the leading coefficient of a PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::MakeSquareFree 
This AdaptableUnaryFunction computes the squarefree part of a polynomial of type PolynomialTraits_d::Polynomial_d up to a constant factor. More...  
concept  PolynomialTraits_d::MonomialRepresentation 
This Functor outputs the monomial representation of the given polynomial, that is, it writes all non zero terms of the polynomial as std::pair<CGAL::Exponent_vector, PolynomialTraits_d::Innermost_coefficient_type> into the given output iterator. More...  
concept  PolynomialTraits_d::Move 
This AdaptableFunctor moves a variable at position \( i\) to a new position \( j\). The relative order of the other variables is preserved, that is, the variables between \( x_i\) and \( x_j\) (including \( x_j\)) are moved by one position while \( x_i\) is moved to the former position of \( x_j\). More...  
concept  PolynomialTraits_d::MultivariateContent 
This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the symmetric view on the polynomial, that is, it computes the gcd of all innermost coefficients. More...  
concept  PolynomialTraits_d::Negate 
This AdaptableUnaryFunction computes \( p(x)\) for a given polynomial \( p\). More...  
concept  PolynomialTraits_d::Permute 
This AdaptableFunctor permutes the variables of the given polynomial with respect to a permutation \( \sigma\), that is, each monomial \( \prod x_i^{e_i}\) will be mapped to the monomial \( \prod x_{\sigma(i)}^{e_i}\). The permutation \( \sigma\) is given by the iterator range of length PolynomialTraits_d::d , which is supposed to contain the second row of the permutation. More...  
concept  PolynomialTraits_d::PolynomialSubresultants 
Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of type PolynomialTraits_d::Polynomial_d with respect to outermost variable. Let \( p=\ccSum{i=0,\ldots,n}{} p_i x^i\) and \( q=\ccSum{i=0,\ldots,m}{} q_i x^i\), where \( x\) is the outermost variable. The \( i\)th subresultant (with \( i=0,\ldots,\min\{n,m\}\)) is defined by. More...  
concept  PolynomialTraits_d::PolynomialSubresultantsWithCofactors 
Computes the polynomial subresultant of two polynomials \( p\) and \( q\) of degree \( n\) and \( m\), respectively, as defined in the documentation of PolynomialTraits_d::PolynomialSubresultants . Moreover, for \( \mathrm{Sres}_i(p,q)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq mi1\) and \( \deg v_i\leq ni1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i p + v_i q\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Sres}_i(p,q)\). More...  
concept  PolynomialTraits_d::PrincipalSturmHabichtSequence 
Computes the principal leading coefficients of the SturmHabicht sequence of a polynomials \( f\) of type PolynomialTraits_d::Polynomial_d with respect a certain variable \( x_i\). This means that for the \( j\)th SturmHabicht polynomial, this methods returns the coefficient of \( x_i^j\). More...  
concept  PolynomialTraits_d::PrincipalSubresultants 
Computes the principal subresultant of two polynomials \( p\) and \( q\) of type PolynomialTraits_d::Coefficient_type with respect to the outermost variable. The \( i\)th principal subresultant, \( \mathrm{sres}_i(p,q)\), is defined as the coefficient at \( t^i\) of the \( i\)th polynomial subresultant \( \mathrm{Sres}_i(p,q)\). Thus, it is either the leading coefficient of \( \mathrm{Sres}_i\), or zero in the case where its degree is below \( i\). More...  
concept  PolynomialTraits_d::PseudoDivision 
This AdaptableFunctor computes the pseudo division of two polynomials \( f\) and \( g\). More...  
concept  PolynomialTraits_d::PseudoDivisionQuotient 
This AdaptableBinaryFunction computes the quotient of the pseudo division of two polynomials \( f\) and \( g\). More...  
concept  PolynomialTraits_d::PseudoDivisionRemainder 
This AdaptableBinaryFunction computes the remainder of the pseudo division of two polynomials \( f\) and \( g\). More...  
concept  PolynomialTraits_d::Resultant 
This AdaptableBinaryFunction computes the resultant of two polynomials \( f\) and \( g\) of type PolynomialTraits_d::Polynomial_d with respect to a certain variable. More...  
concept  PolynomialTraits_d::Scale 
Given a constant \( c\) this AdaptableBinaryFunction scales a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable, that is, it computes \( p(c\cdot x)\). More...  
concept  PolynomialTraits_d::ScaleHomogeneous 
Given a numerator \( a\) and a denominator \( b\) this AdaptableFunctor scales a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable, that is, it computes \( b^{degree(p)}\cdot p(a/b\cdot x)\). More...  
concept  PolynomialTraits_d::Shift 
This AdaptableBinaryFunction multiplies a PolynomialTraits_d::Polynomial_d by the given power of the specified variable. More...  
concept  PolynomialTraits_d::SignAt 
This AdaptableFunctor returns the sign of a PolynomialTraits_d::Polynomial_d \( p\) at given Cartesian point represented as an iterator range. More...  
concept  PolynomialTraits_d::SignAtHomogeneous 
This AdaptableFunctor returns the sign of a PolynomialTraits_d::Polynomial_d \( p\) at a given homogeneous point, which is given by an iterator range. More...  
concept  PolynomialTraits_d::SquareFreeFactorize 
This Functor computes a squarefree factorization of a PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor 
This AdaptableFunctor computes a squarefree factorization up to a constant factor (utcf) of a PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::SturmHabichtSequence 
Computes the SturmHabicht sequence (aka the signed subresultant sequence) of a polynomial \( f\) of type PolynomialTraits_d::Polynomial_d with respect to a certain variable \( x_i\). The SturmHabicht sequence is similar to the polynomial subresultant sequence of \( f\) and its derivative \( f':=\frac{\partial f}{\partial x_i}\) with respect to \( x_i\). The implementation is based on the following definition: More...  
concept  PolynomialTraits_d::SturmHabichtSequenceWithCofactors 
Computes the SturmHabicht polynomials of a polynomial \( f\) of degree \( n\), as defined in the documentation of PolynomialTraits_d::SturmHabichtSequence . Moreover, for \( \mathrm{Stha}_i(f)\), polynomials \( u_i\) and \( v_i\) with \( \deg u_i\leq ni2\) and \( \deg v_i\leq ni1\) are computed such that \( \mathrm{Sres}_i(p,q)=u_i f + v_i f'\). \( u_i\) and \( v_i\) are called the cofactors of \( \mathrm{Stha}_i(f)\). More...  
concept  PolynomialTraits_d::Substitute 
This Functor substitutes all variables of a given multivariate PolynomialTraits_d::Polynomial_d by the values given in the iterator range, where begin refers the value for the innermost variable. More...  
concept  PolynomialTraits_d::SubstituteHomogeneous 
This Functor substitutes all variables of a given multivariate PolynomialTraits_d::Polynomial_d \( p\) by the values given in the iterator range, where begin refers the value for the innermost variable. In contrast to PolynomialTraits_d::Substitute the given polynomial \( p\) is interpreted as a homogeneous polynomial. Hence the iterator range is required to be of length PolynomialTraits_d::d+1 . More...  
concept  PolynomialTraits_d::Swap 
This AdaptableFunctor swaps two variables of a multivariate polynomial. More...  
concept  PolynomialTraits_d::TotalDegree 
This AdaptableUnaryFunction computes the total degree of a PolynomialTraits_d::Polynomial_d . More...  
concept  PolynomialTraits_d::Translate 
This AdaptableBinaryFunction translates a PolynomialTraits_d::Polynomial_d with respect to one variable, that is, for a given polynomial \( p\) and constant \( c\) it computes \( p(x+c)\). More...  
concept  PolynomialTraits_d::TranslateHomogeneous 
Given numerator \( a\) and denominator \( b\) this AdaptableFunctor translates a PolynomialTraits_d::Polynomial_d \( p\) with respect to one variable by \( a/b\), that is, it computes \( b^{degree(p)}\cdot p(x+a/b)\). More...  
concept  PolynomialTraits_d::UnivariateContent 
This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the univariate (recursive) view on the polynomial, that is, it computes the gcd of all coefficients with respect to one variable. More...  
concept  PolynomialTraits_d::UnivariateContentUpToConstantFactor 
This AdaptableUnaryFunction computes the content of a PolynomialTraits_d::Polynomial_d with respect to the univariate (recursive) view on the polynomial up to a constant factor (utcf), that is, it computes the \( \mathrm{gcd\_utcf}\) of all coefficients with respect to one variable. More...  
concept  PolynomialTraits_d 
A model of PolynomialTraits_d is associated with a type Polynomial_d . The type Polynomial_d represents a multivariate polynomial. The number of variables is denoted as the dimension \( d\) of the polynomial, it is arbitrary but fixed for a certain model of this concept. Note that univariate polynomials are not excluded by this concept. In this case \( d\) is just set to one. More...  