mirror of https://github.com/CGAL/cgal
61 lines
2.1 KiB
TeX
61 lines
2.1 KiB
TeX
\begin{ccRefConcept}{PolynomialTraits_d::Resultant}
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\ccDefinition
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This \ccc{AdaptableBinaryFunction} computes the resultant of two polynomials
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$f$ and $g$ of type \ccc{PolynomialTraits_d::Polynomial_d} with respect 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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Given two polynomials $f$ and $g$ over some \ccc{IntegralDomain}, where
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\[f := f_n \prod_{i=1}^{n}{(x-\alpha_i)}\]
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and
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\[g := g_n \prod_{j=1}^{m}{(x-\beta_j)}.\]
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The resultant of $f$ and $g$ is defined as
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\[f_n g_n \prod_{i=1}^{n}\prod_{i=1}^{m}{(\alpha_i-\beta_j)}.\]
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There are various ways to compute the resultant.
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Naive options are the computation of the
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resultant as the determinant of 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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\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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\end{ccRefConcept}
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