\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.5 - Polynomial
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PolynomialTraits_d::GcdUpToConstantFactor Concept Reference
Concepts

Definition

This AdaptableBinaryFunction computes the \( gcd\) up to a constant factor (utcf) of two polynomials of type PolynomialTraits_d::Polynomial_d.

In case the base ring \( R\) (PolynomialTraits_d::Innermost_coefficient_type) is not a UniqueFactorizationDomain or not a Field the polynomial ring \( R[x_0,\dots,x_{d-1}]\) (PolynomialTraits_d::Polynomial_d) may not possesses greatest common divisors. However, since \( R\) is an integral domain one can consider its quotient field \( Q(R)\) for which \( gcd\)s of polynomials exist.

This functor computes \( gcd\_utcf(f,g) = D * gcd(f,g)\), for some \( D \in R\) such that \( gcd\_utcf(f,g) \in R[x_0,\dots,x_{d-1}]\). Hence, \( gcd\_utcf(f,g)\) may not be a divisor of \( f\) and \( g\) in \( R[x_0,\dots,x_{d-1}]\).

Refines:

AdaptableBinaryFunction

CopyConstructible

DefaultConstructible

See Also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::IntegralDivisionUpToConstantFactor
PolynomialTraits_d::UnivariateContentUpToConstantFactor
PolynomialTraits_d::SquareFreeFactorizeUpToConstantFactor

Types

typedef
PolynomialTraits_d::Polynomial_d 		result_type
 
typedef
PolynomialTraits_d::Polynomial_d 		first_argument_type
 
typedef
PolynomialTraits_d::Polynomial_d 		second_argument_type
 

Operations

result_type operator() (first_argument_type f, second_argument_type g)
 Computes \( gcd(f,g)\) up to a constant factor.