mirror of https://github.com/CGAL/cgal
107 lines
3.7 KiB
TeX
107 lines
3.7 KiB
TeX
\begin{ccRefConcept}{PolynomialTraits_d::Resultant}
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\ccDefinition
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This \ccc{AdaptableBinaryFunction} computes the {\em resultant} of two polynomials
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$f$ and $g$ of type \ccc{PolynomialTraits_d::Polynomial_d} with respect to a
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certain variable.
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Note that this functor operates on the polynomial in the univariate view,
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that is, the polynomial is considered as a univariate polynomial in one
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specific variable.
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Let $f$ and $g$ be two univariate polynomials over some commutative ring $A$,
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where
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\[f = f_mx^m + \dots + f_0\] and
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\[g = g_nx^n + \dots + g_0.\]
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The resultant of $f$ and $g$ is defined as the determinant of the {\em Sylvester matrix}:
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\begin{ccTexOnly}
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\[ \left( \begin{array}{cccccccc}
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f_m & \dots & f_0&&& \\
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&f_m & \dots & f_0&&\\
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&&\ddots&&\ddots&\\
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&&&f_m &\dots& f_0\\
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g_n & \dots & g_0&&& \\
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&g_n & \dots & g_0&&\\
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&&\ddots&&\ddots&\\
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&&&g_n & \dots& g_0\\
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\end{array} \right) \]
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\end{ccTexOnly}
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\begin{ccHtmlOnly}
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<CENTER>
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<IMG BORDER=0 SRC="./sylvester_matrix.gif" ALIGN=middle ALT="Sylvester Matrix">
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</CENTER>
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\end{ccHtmlOnly}
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Note that this is a $(n+m)\times(n+m)$ matrix as there are $n$ rows for $f$
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and $m$ rows that are used for $g$. The blank spaces are supposed to be
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filled with zeros.
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% In the sequel we list the most important properties of the resultant.
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\begin{ccAdvanced}
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Let $L$ be the algebraic closure of $A$, and write $f$ and $g$ as
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\[f := f_m \ccProd{i=1}{m}{(x-\alpha_i)},\ \alpha_i \in L \]
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and
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\[g := g_n \ccProd{j=1}{n}{(x-\beta_j)},\ \beta_i \in L,\] then
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the resultant of $f$ and $g$ is (up to leading coefficients)
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the product of all pairwise differences of the roots of $f$ and $g$, namely
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\[res(f,g) = f_m^n g_n^m \ccProd{i=1}{m}{\ccProd{j=1}{n}{(\alpha_i-\beta_j)}}.\]
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In particular, $res(f,g) \neq 0$ iff $f$ and $g$ have a common factor with a
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positive degree in $X$. %, that is, they may still have a common factor in $A$.
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%As a consequence $res(f,f') \neq 0$ iff $f$ is square free with respect to
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%$X$, that is, the content of $f$ may still contain a square.
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\end{ccAdvanced}
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There are various ways to compute the resultant.
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Naive options are the computation of the resultant as the determinant of
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the Sylvester Matrix or the Bezout
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Matrix as well as the so called subresultant algorithm,
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which is a variant of the Euclidean Algorithm.
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More sophisticated methods may use modular arithmetic and interpolation.
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For more information we refer to, e.g., \cite{gg-mca-99}.
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\ccRefines
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\ccc{AdaptableBinaryFunction}\\
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\ccc{CopyConstructible}\\
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\ccc{DefaultConstructible}\\
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\ccTypes
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\ccSetThreeColumns{xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx}{xxx}{}
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\ccCreationVariable{fo}
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\ccTypedef{typedef PolynomialTraits_d::Coefficient_type result_type;}{}
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\ccGlue
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\ccTypedef{typedef PolynomialTraits_d::Polynomial_d first_argument_type;}{}
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\ccGlue
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\ccTypedef{typedef PolynomialTraits_d::Polynomial_d second_argument_type;}{}
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\ccOperations
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\ccMethod{result_type operator()(first_argument_type f,
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second_argument_type g);}
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{ Computes the resultant of $f$ and $g$,
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with respect to the outermost variable.}
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%\ccMethod{result_type operator()(first_argument_type f,
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% second_argument_type g,
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% int i);}
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% { Computes the resultant of $f$ and $g$,
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% with respect to variable $x_i$.
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% \ccPrecond $0 \leq i < d$
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% }
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%\ccHasModels
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\ccSeeAlso
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\ccRefIdfierPage{Polynomial_d}\\
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\ccRefIdfierPage{PolynomialTraits_d}\\
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\ccRefIdfierPage{PolynomialTraits_d::UnivariateContent}\\
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\ccRefIdfierPage{PolynomialTraits_d::PolynomialSubresultants}\\
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\ccRefIdfierPage{PolynomialTraits_d::PrincipalSubresultants}\\
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\end{ccRefConcept}
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