Sylvester determinant

E571003

determinant mathematical concept

The Sylvester determinant is a mathematical construct introduced by James Joseph Sylvester, typically referring to a determinant associated with resultants and elimination theory in algebra.

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Statements (41)

Predicate	Object
instanceOf	determinant ⓘ mathematical concept ⓘ
appearsIn	classical invariant theory ⓘ theory of polynomial equations ⓘ
appliesTo	systems of polynomial equations ⓘ univariate polynomials ⓘ
category	determinants in linear algebra ⓘ resultant theory ⓘ
condition	nonzero value implies no common root between the polynomials in an algebraic closure ⓘ
constructedFrom	coefficients of the first polynomial arranged in shifted rows ⓘ coefficients of the second polynomial arranged in shifted rows ⓘ
definedAs	the determinant of the Sylvester matrix of two polynomials ⓘ
dependsOn	coefficients of the given polynomials ⓘ
field	algebra ⓘ algebraic geometry ⓘ computational algebra ⓘ elimination theory NERFINISHED ⓘ
generalizationOf	resultant of two univariate polynomials ⓘ
historicalNote	introduced in the 19th century by James Joseph Sylvester ⓘ
matrixSize	(m+n)×(m+n) for polynomials of degrees m and n ⓘ
namedAfter	James Joseph Sylvester NERFINISHED ⓘ
namedEntityType	mathematical object ⓘ
property	is a polynomial in the coefficients of the input polynomials ⓘ is homogeneous in the coefficients of each polynomial ⓘ vanishes if and only if the polynomials have a common root in an algebraic closure ⓘ
relatedTo	Bezout matrix ⓘ Groebner basis methods (as an alternative elimination tool) ⓘ Sylvester matrix NERFINISHED ⓘ Sylvester resultant NERFINISHED ⓘ discriminant of a polynomial ⓘ resultant ⓘ
symbol	often denoted as the determinant of the Sylvester matrix S(f,g) ⓘ
usedFor	computing the resultant of polynomials ⓘ deriving algebraic conditions for intersection of plane curves ⓘ elimination of variables ⓘ implicitization of parametrically defined curves or surfaces ⓘ testing whether two polynomials have a common root ⓘ
usedIn	algebraic elimination algorithms ⓘ computational algebraic geometry ⓘ computer algebra systems ⓘ symbolic computation ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

James Joseph Sylvester → notableWork → Sylvester determinant ⓘ