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

CGAL 4.8 - Polynomial

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Polynomial/subresultants.cpp
#include <CGAL/Polynomial.h>
#include <CGAL/Polynomial_traits_d.h>
#include <CGAL/Polynomial_type_generator.h>
#include <CGAL/Exact_integer.h>
int main(){
CGAL::set_pretty_mode(std::cout);
typedef CGAL::Exact_integer Int;
typedef CGAL::Polynomial_type_generator<Int,1>::Type Poly_1;
typedef CGAL::Polynomial_traits_d<Poly_1> PT_1;
//construction using shift
Poly_1 x = PT_1::Shift()(Poly_1(1),1); // x^1
Poly_1 F // = (x+1)^2*(x-1)*(2x-1)=2x^4+x^3-3x^2-x+1
= 2 * CGAL::ipower(x,4) + 1 * CGAL::ipower(x,3)
- 3 * CGAL::ipower(x,2) - 1 * CGAL::ipower(x,1)
+ 1 * CGAL::ipower(x,0);
std::cout << "F=" << F << std::endl;
Poly_1 G // = (x+1)*(x+3)=x^2+4*x+3
= 1 * CGAL::ipower(x,2) + 4 * CGAL::ipower(x,1) + 3 * CGAL::ipower(x,0);
std::cout << "G=" << G << std::endl;
// Resultant computation:
PT_1::Resultant resultant;
std::cout << "The resultant of F and G is: " << resultant(F,G) << std::endl;
// It is zero, because F and G have a common factor
// Real root counting:
PT_1::Principal_sturm_habicht_sequence stha;
std::vector<Int> psc;
stha(F,std::back_inserter(psc));
int roots = CGAL::number_of_real_roots(psc.begin(),psc.end());
std::cout << "The number of real roots of F is: " << roots << std::endl; // 3
roots = CGAL::number_of_real_roots(G);
std::cout << "The number of real roots of G is: " << roots << std::endl; // 2
return 0;
}