\begin{ccRefConcept}{PolynomialTraits_d}

\ccDefinition
A model of \ccc{PolynomialTraits_d} is associated to an type 
\ccc{Polynomial_d}, representing a multivariate polynomial. 
The number of variables is denoted as the dimension of the polynomial,
it is arbitrary but fixed for a certain model of this concept.  

Note: That this concept does not exclude univariate polynomial. 


\ccc{PolynomialTraits_d} provides two different views on the 
multivariate polynomial. 

\begin{itemize}
\item A recursive view, that sees the polynomial as an element of 
$R[x_1,\dots,x_{d-1}][x_d]$. In this view, the polynomial is handled as
an univariate polynomial over the ring $R[x_1,\dots,x_{d-1}]$. 
\item A symmetric view, which is symmetric with respect to all variables,
seeing the polynomials as element of $R[x_1,\dots,x_d]$.
\end{itemize}


The default view is the recursive view, therefore all functors are 
designed such that there default version performs the operation 
with respect to this view. 

\ccRefines

\ccConstants
 
\ccVariable{const int d;}{The dimension and the number of variables respectively.}

\ccTypes

\ccNestedType{Polynomial_d}{ Type representing $R[x_1,\dots,x_{d}]$.}\ccGlue
\ccNestedType{Coefficient }{ Type representing $R[x_1,\dots,x_{d-1}]$.}\ccGlue
\ccNestedType{Innermost_coefficient}{ Type representing the base ring $R$.}

\ccHeading{Functors}

\ccSetTwoColumns{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}{}
\ccNestedType{Construct_polynomial_d}{ A model of \ccc{PolynomialTraits_d::ConstructPolynomial_d}}

%Properties
\ccNestedType{Degree}{ A model of \ccc{PolynomialTraits_d::Degree}}
\ccNestedType{Total_degree}{ A model of \ccc{PolynomialTraits_d::TotalDegree}}
\ccNestedType{Leading_coefficient}{ A model of \ccc{PolynomialTraits_d::LeadingCoefficient}}

\ccNestedType{Univariate_content}{ 
        In case \ccc{PolynomialTraits_d::Coefficient} is {\bf not} a model of 
        \ccc{UFDomain}, this is \ccc{CGAL::Null_type}, otherwise this is 
        a model of \ccc{PolynomialTraits_d::UnivariateContent}}
\begin{ccAdvanced}
\ccNestedType{Multivariate_content}{ 
        In case \ccc{PolynomialTraits_d::Innermost_coefficient} is {\bf not} 
        a model of \ccc{UFDomain}, this is \ccc{CGAL::Null_type}, 
        otherwise this is a model of 
        \ccc{PolynomialTraits_d::MultivariateContent}}
\end{ccAdvanced}

%Manipulation
\ccNestedType{Shift}{ A model of \ccc{PolynomialTraits_d::Shift}}
\ccNestedType{Negate}{ A model of \ccc{PolynomialTraits_d::Negate}}
\ccNestedType{Invert}{ A model of \ccc{PolynomialTraits_d::Invert}}

\ccNestedType{Translate}{ A model of \ccc{PolynomialTraits_d::Translate}}
\ccNestedType{Translate_homogeneous}{ A model of \ccc{PolynomialTraits_d::TranslateHomogeneous}}

\ccNestedType{Scale}{ A model of \ccc{PolynomialTraits_d::Scale}}
\ccNestedType{Scale_homogeneous}{ A model of \ccc{PolynomialTraits_d::ScaleHomogeneous}}
\begin{ccAdvanced}
//\ccNestedType{Scale_up}{ A model of \ccc{PolynomialTraits_d::ScaleUp, return $p(a*x)$}}
//\ccNestedType{Scale_down}{ A model of \ccc{PolynomialTraits_d::ScaleDown, return $b^{degree}*p(x/b)$}}
\end{ccAdvanced}

%unary operations
\ccNestedType{Differentiate}{ A model of \ccc{PolynomialTraits_d::Differentiate}}
\ccNestedType{Make_square_free}{ A model of \ccc{PolynomialTraits_d::MakeSquareFree}}

\ccNestedType{Square_free_factorization}{ In case \ccc{PolynomialTraits::Polynomial_d} 
        is not a model of \ccc{UFDomain}, this is of type \ccc{CGAL::Null_type},
        otherwise this is a model of \ccc{PolynomialTraits_d::SquareFreeFactorization}}


%pseudo division
\ccNestedType{Pseudo_division          }{ A model of \ccc{PolynomialTraits_d::Pseudo_division}}
\ccNestedType{Pseudo_division_remainder}{ A model of \ccc{PolynomialTraits_d::Pseudo_division_remainder}}
\ccNestedType{Pseudo_division_quotient }{ A model of \ccc{PolynomialTraits_d::Pseudo_division_quotient}}

%utcf
\ccNestedType{Gcd_up_to_constant_factor}{ A model of \ccc{PolynomialTraits_d::GcdUpToConstantFactor}}
\ccNestedType{Integral_division_up_to_constant_factor}{ A model of \ccc{PolynomialTraits_d::IntegralDivisionUpToConstantFactor}}
\ccNestedType{Content_up_to_constant_factor}{ A model of \ccc{PolynomialTraits_d::ContentUpToConstantFactor}}
\ccNestedType{Square_free_factorization_up_to_constant_factor}{ A model of \ccc{PolynomialTraits_d::SquareFreeFactorizationUpToConstantFactor}}

%Evaluation
\ccNestedType{Evaluate}{ A model of \ccc{PolynomialTraits_d::Evaluate}}
\ccNestedType{Evaluate_homogeneous}{ A model of \ccc{PolynomialTraits_d::EvaluateHomogeneous}}
%\ccNestedType{Sign_at}{ A model of \ccc{PolynomialTraits_d::SignAt}}


%resultant
\ccNestedType{Resultant}{ A model of \ccc{PolynomialTraits_d::Resultant}}




\ccSeeAlso 

\ccRefIdfierPage{AlgebraicStructureTraits}\\
\ccRefIdfierPage{Polynomial_d}\\

\end{ccRefConcept}