Sylvester determinant
E571003
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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Sylvester determinant canonical | 2 |
| Sylvester resultant | 2 |
| Sylvester’s determinant identity | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6149921 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Sylvester determinant Context triple: [James Joseph Sylvester, notableWork, Sylvester determinant]
-
A.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
B.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
C.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
-
D.
Toeplitz matrices
Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
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E.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Sylvester determinant Target entity description: 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.
-
A.
Cauchy determinant
The Cauchy determinant is a classical determinant formula in linear algebra that gives a closed-form expression for matrices with entries of the form 1/(x_i + y_j), named after the French mathematician Augustin-Louis Cauchy.
-
B.
Cauchy matrix
A Cauchy matrix is a structured matrix whose entries are defined by the reciprocals of pairwise differences of two sequences, widely used in numerical analysis, interpolation, and algebra.
-
C.
Cauchy interlacing theorem
The Cauchy interlacing theorem is a fundamental result in linear algebra that relates the eigenvalues of a symmetric matrix to those of its principal submatrices, showing how they "interlace" on the real line.
-
D.
Toeplitz matrices
Toeplitz matrices are structured matrices whose entries are constant along each diagonal, playing a central role in operator theory, numerical analysis, and signal processing.
-
E.
Clebsch–Aronhold invariants
The Clebsch–Aronhold invariants are classical algebraic invariants associated with binary forms, particularly quartic forms, that play a key role in invariant theory and the classification of algebraic curves.
- F. None of above. chosen
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 ⓘ |
| 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 ⓘ |
| 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 ⓘ Sylvester resultant ⓘ
linked to:
Sylvester determinant
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Sylvester determinant Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject linked to:
Sylvester
linked to: Sylvester determinant
linked to: Sylvester determinant
linked to: Sylvester determinant