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

CGAL 4.8 - Polynomial

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Polynomial/substitute.cpp
#include <CGAL/Polynomial.h>
#include <CGAL/Polynomial_traits_d.h>
#include <CGAL/Polynomial_type_generator.h>
int main(){
CGAL::set_pretty_mode(std::cout);
typedef CGAL::Polynomial_type_generator<int,2>::Type Poly_2;
typedef CGAL::Polynomial_traits_d<Poly_2> PT_2;
//construction using shift
Poly_2 x = PT_2::Shift()(Poly_2(1),1,0); // x^1
Poly_2 y = PT_2::Shift()(Poly_2(1),1,1); // y^1
Poly_2 F = 2*x*y + 3*CGAL::ipower(y,3);
std::cout << "The bivariate polynomial F: " << F // = 3*y^3 + (2*x)*y
<< std::endl << std::endl;
PT_2::Evaluate evaluate;
PT_2::Evaluate_homogeneous hevaluate;
// Evaluation considers a polynomials as univariate:
std::cout << "F(5): " << evaluate(F,5) // = 10*x + 375
<< std::endl;
// Evaluate_homogeneous considers F as a homogeneous polynomial in
// the outermost variable only, that is, F is interpreted as
// F(u,v) = 2*x*u*v^2 + 3 * u^3
std::cout << "F(5,7): " << hevaluate(F,5,7) // = 490*x + 375
<< std::endl << std::endl;
PT_2::Substitute substitute;
PT_2::Substitute_homogeneous hsubstitute;
// Substitute considers a polynomials as multivariate, that is, the
// new values for the variables are given by an iterator range
// Note that the value type only has to be interoperable with the innermost
// coefficient
std::list<Poly_2> replacements;
replacements.push_back(x-1); // replace x by x-1
replacements.push_back(y); // replace y by y, i.e., do nothing
std::cout << "The bivariate polynomial F: " << F // = 3*y^3 + (2*x)*y
<< std::endl;
std::cout << "F(x-1,y): " // = 3*y^3 + (2*x + (-2))*y
<< substitute(F,replacements.begin(),replacements.end())
<< std::endl;
// Substitute_homogeneous considers F as a homogeneous polynomial in
// all variable, that is, F is interpreted as
// F(x,y,w) = 2*x*y*w + 3 * y^3
replacements.push_back(y); // replace z by y
std::cout << "F(x-1,y,y): " // = 3*y^3 + (2*x + (-2))*y^2
<< hsubstitute(F,replacements.begin(),replacements.end())
<< std::endl;
}