"Fossies" - the Fresh Open Source Software Archive

Member "doc_html/Polynomial/classPolynomialTraits__d_1_1Resultant.html" (8 Nov 2019, 14900 Bytes) of package /linux/misc/CGAL-4.14.2-doc_html.tar.xz:


Caution: In this restricted "Fossies" environment the current HTML page may not be correctly presentated and may have some non-functional links. You can here alternatively try to browse the pure source code or just view or download the uninterpreted raw source code. If the rendering is insufficient you may try to find and view the page on the CGAL-4.14.2-doc_html.tar.xz project site itself.

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

CGAL 4.14.2 - Polynomial
PolynomialTraits_d::Resultant Concept Reference

Definition

This AdaptableBinaryFunction computes the resultant of two polynomials \( f\) and \( g\) of type PolynomialTraits_d::Polynomial_d with respect to a certain variable.

Note that this functor operates on the polynomial in the univariate view, that is, the polynomial is considered as a univariate polynomial in one specific variable.

Let \( f\) and \( g\) be two univariate polynomials over some commutative ring \( A\), where

\[ f = f_mx^m + \dots + f_0 \]

and

\[ g = g_nx^n + \dots + g_0. \]

The resultant of \( f\) and \( g\) is defined as the determinant of the Sylvester matrix:

sylvester_matrix.png

Note that this is a \( (n+m)\times(n+m)\) matrix as there are \( n\) rows for \( f\) and \( m\) rows that are used for \( g\). The blank spaces are supposed to be filled with zeros.

Advanced

Let \( L\) be the algebraic closure of \( A\), and write \( f\) and \( g\) as

\[ f := f_m \ccProd{i=1}{m}{(x-\alpha_i)},\ \alpha_i \in L \]

and

\[ g := g_n \ccProd{j=1}{n}{(x-\beta_j)},\ \beta_i \in L, \]

then the resultant of \( f\) and \( g\) is (up to leading coefficients) the product of all pairwise differences of the roots of \( f\) and \( g\), namely

\[ res(f,g) = f_m^n g_n^m \ccProd{i=1}{m}{\ccProd{j=1}{n}{(\alpha_i-\beta_j)}}. \]

In particular, \( res(f,g) \neq 0\) iff \( f\) and \( g\) have a common factor with a positive degree in \( X\).

There are various ways to compute the resultant. Naive options are the computation of the resultant as the determinant of the Sylvester Matrix or the Bezout Matrix as well as the so called subresultant algorithm, which is a variant of the Euclidean Algorithm. More sophisticated methods may use modular arithmetic and interpolation. For more information we refer to, e.g., [2].

Refines:

AdaptableBinaryFunction

CopyConstructible

DefaultConstructible

See also
Polynomial_d
PolynomialTraits_d
PolynomialTraits_d::UnivariateContent
PolynomialTraits_d::PolynomialSubresultants
PolynomialTraits_d::PrincipalSubresultants

Types

typedef PolynomialTraits_d::Coefficient_type 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 the resultant of \( f\) and \( g\), with respect to the outermost variable.