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- Polynomial_greatest_common_divisor abstract "In algebra, the greatest common divisor (frequently abbreviated as GCD) of two polynomials is a polynomial, of the highest possible degree, that is a factor of both the two original polynomials. This concept is analogous to the greatest common divisor of two integers.In the important case of univariate polynomials over a field the polynomial GCD may be computed, like for the integer GCD, by Euclid's algorithm using long division. The polynomial GCD is defined only up to the multiplication by an invertible constant.The similarity between the integer GCD and the polynomial GCD allows us to extend to univariate polynomials all the properties that may be deduced from Euclid's algorithm and Euclidean division. Moreover, the polynomial GCD has specific properties that make it a fundamental notion in various areas of algebra. Typically, the roots of the GCD of two polynomials are the common roots of the two polynomials, and this allows to get information on the roots without computing them. For example, the multiple roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow to compute the square-free factorization of the polynomial, which provides polynomials whose roots are the roots of a given multiplicity.The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain. There exist algorithms to compute them as soon as one has a GCD algorithm in the ring of coefficients. These algorithms proceed by a recursion on the number of variables to reduce the problem to a variant of Euclid's algorithm. They are a fundamental tool in computer algebra, because computer algebra systems use them systematically to simplify fractions. Conversely, most of the modern theory of polynomial GCD has been developed to satisfy the need of efficiency of computer algebra systems.".
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- Polynomial_greatest_common_divisor wikiPageWikiLink Algebraic_extension.
- Polynomial_greatest_common_divisor wikiPageWikiLink Bxc3xa9zouts_identity.
- Polynomial_greatest_common_divisor wikiPageWikiLink Category:Computer_algebra.
- Polynomial_greatest_common_divisor wikiPageWikiLink Category:Polynomials.
- Polynomial_greatest_common_divisor wikiPageWikiLink Chinese_remainder_theorem.
- Polynomial_greatest_common_divisor wikiPageWikiLink Coefficient.
- Polynomial_greatest_common_divisor wikiPageWikiLink Coefficients.
- Polynomial_greatest_common_divisor wikiPageWikiLink Computer_algebra.
- Polynomial_greatest_common_divisor wikiPageWikiLink Computer_algebra_system.
- Polynomial_greatest_common_divisor wikiPageWikiLink Coprime.
- Polynomial_greatest_common_divisor wikiPageWikiLink Coprime_integers.
- Polynomial_greatest_common_divisor wikiPageWikiLink Donald_E._Knuth.
- Polynomial_greatest_common_divisor wikiPageWikiLink Donald_Knuth.
- Polynomial_greatest_common_divisor wikiPageWikiLink Euclidean_algorithm.
- Polynomial_greatest_common_divisor wikiPageWikiLink Euclidean_division.
- Polynomial_greatest_common_divisor wikiPageWikiLink Euclidean_domain.
- Polynomial_greatest_common_divisor wikiPageWikiLink Euclids_algorithm.
- Polynomial_greatest_common_divisor wikiPageWikiLink Factorization.
- Polynomial_greatest_common_divisor wikiPageWikiLink Factorization_of_polynomials.
- Polynomial_greatest_common_divisor wikiPageWikiLink Field_(mathematics).
- Polynomial_greatest_common_divisor wikiPageWikiLink Field_of_fractions.
- Polynomial_greatest_common_divisor wikiPageWikiLink Finite_field.
- Polynomial_greatest_common_divisor wikiPageWikiLink Finite_ring.
- Polynomial_greatest_common_divisor wikiPageWikiLink Formal_derivative.
- Polynomial_greatest_common_divisor wikiPageWikiLink Greatest_common_divisor.
- Polynomial_greatest_common_divisor wikiPageWikiLink Gröbner_basis.
- Polynomial_greatest_common_divisor wikiPageWikiLink Hadamard_inequality.
- Polynomial_greatest_common_divisor wikiPageWikiLink Hadamards_inequality.
- Polynomial_greatest_common_divisor wikiPageWikiLink Identity_matrix.
- Polynomial_greatest_common_divisor wikiPageWikiLink Integral_domain.
- Polynomial_greatest_common_divisor wikiPageWikiLink List_of_polynomial_topics.
- Polynomial_greatest_common_divisor wikiPageWikiLink Modular_arithmetic.
- Polynomial_greatest_common_divisor wikiPageWikiLink Monic_polynomial.
- Polynomial_greatest_common_divisor wikiPageWikiLink Multiple_root.
- Polynomial_greatest_common_divisor wikiPageWikiLink Multiplicity_(mathematics).
- Polynomial_greatest_common_divisor wikiPageWikiLink Multivariate_division_algorithm.
- Polynomial_greatest_common_divisor wikiPageWikiLink Multivariate_polynomial.
- Polynomial_greatest_common_divisor wikiPageWikiLink Polynomial.
- Polynomial_greatest_common_divisor wikiPageWikiLink Polynomial_factorization.
- Polynomial_greatest_common_divisor wikiPageWikiLink Polynomial_long_division.
- Polynomial_greatest_common_divisor wikiPageWikiLink Principal_ideal_domain.
- Polynomial_greatest_common_divisor wikiPageWikiLink Rational_reconstruction_(mathematics).
- Polynomial_greatest_common_divisor wikiPageWikiLink Recursion.
- Polynomial_greatest_common_divisor wikiPageWikiLink Resultant.
- Polynomial_greatest_common_divisor wikiPageWikiLink Root-finding_algorithm.
- Polynomial_greatest_common_divisor wikiPageWikiLink Root_of_a_function.
- Polynomial_greatest_common_divisor wikiPageWikiLink Singular_value_decomposition.
- Polynomial_greatest_common_divisor wikiPageWikiLink Square-free_factorization.
- Polynomial_greatest_common_divisor wikiPageWikiLink Square-free_polynomial.
- Polynomial_greatest_common_divisor wikiPageWikiLink Sturm_sequence.
- Polynomial_greatest_common_divisor wikiPageWikiLink Sturms_theorem.
- Polynomial_greatest_common_divisor wikiPageWikiLink Sylvester_matrix.
- Polynomial_greatest_common_divisor wikiPageWikiLink Symbolic_computation.
- Polynomial_greatest_common_divisor wikiPageWikiLink Synthetic_division.
- Polynomial_greatest_common_divisor wikiPageWikiLink Total_order.
- Polynomial_greatest_common_divisor wikiPageWikiLink Unique_factorization_domain.
- Polynomial_greatest_common_divisor wikiPageWikiLink Unit_(ring_theory).
- Polynomial_greatest_common_divisor wikiPageWikiLink Univariate.
- Polynomial_greatest_common_divisor wikiPageWikiLink Up_to.
- Polynomial_greatest_common_divisor wikiPageWikiLink Zero_of_a_function.
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Arithmetic of algebraic extensions".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Bézout's identity for polynomials".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Euclid's algorithm".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Euclidean division".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Extended Euclidean algorithm".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "GCD".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor#Euclid's algorithm".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor#Euclidean division ".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor#Euclidean division".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor#Pseudo-remainder sequences".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "Polynomial greatest common divisor#Subresultants".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "greatest common divisor (GCD)".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "greatest common divisor".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "highest common factor".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "polynomial greatest common divisor".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "polynomial greatest common divisor#Bézout's identity and extended GCD algorithm".
- Polynomial_greatest_common_divisor wikiPageWikiLinkText "subresultant pseudo-remainder sequence".
- Polynomial_greatest_common_divisor hasPhotoCollection Polynomial_greatest_common_divisor.
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- Polynomial_greatest_common_divisor wikiPageUsesTemplate Template:Polynomials.
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- Polynomial_greatest_common_divisor wikiPageUsesTemplate Template:See_also.
- Polynomial_greatest_common_divisor subject Category:Computer_algebra.
- Polynomial_greatest_common_divisor subject Category:Polynomials.
- Polynomial_greatest_common_divisor hypernym Polynomial.
- Polynomial_greatest_common_divisor type Article.
- Polynomial_greatest_common_divisor type Type.
- Polynomial_greatest_common_divisor type Algorithm.
- Polynomial_greatest_common_divisor type Article.