Clebsch–Aronhold invariants
E262451
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Clebsch–Aronhold invariants canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T2408491 — 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: Clebsch–Aronhold invariants Context triple: [Alfred Clebsch, notableWork, Clebsch–Aronhold invariants]
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A.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
B.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
C.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.
- 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: Clebsch–Aronhold invariants Target entity description: 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.
-
A.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
B.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
C.
The Classical Groups: Their Invariants and Representations
The Classical Groups: Their Invariants and Representations is a foundational mathematical monograph by Hermann Weyl that systematically develops the theory of classical Lie groups, their invariants, and their representation theory.
-
D.
Über die Bildung des Formensystems der ternären biquadratischen Form
"Über die Bildung des Formensystems der ternären biquadratischen Form" is the 1907 doctoral dissertation of mathematician Emmy Noether, in which she investigates the invariant theory of certain higher-degree algebraic forms.
-
E.
Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)
Noether’s AF+BG theorem is a foundational result in algebraic geometry that provides conditions under which a polynomial vanishing on the intersection of two plane curves can be expressed as a linear combination of their defining equations.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
classical invariant ⓘ object in invariant theory ⓘ |
| appliesTo |
binary forms
ⓘ
binary quartic forms ⓘ |
| associatedWith |
binary quartic representation of four-point configurations on P1
ⓘ
genus-one curves represented as double covers of the projective line ⓘ j-invariant of elliptic curves ⓘ |
| assumes | base field of characteristic not equal to 2 or 3 ⓘ |
| context |
19th-century German school of algebra
ⓘ
classical theory of binary forms ⓘ |
| degree |
2
ⓘ
3 ⓘ |
| dependsOn | coefficients of a binary quartic form ⓘ |
| field |
algebraic geometry
ⓘ
classical algebra ⓘ invariant theory ⓘ |
| hasComponent |
cubic invariant of a binary quartic
ⓘ
quadratic invariant of a binary quartic ⓘ |
| introducedIn | 19th century ⓘ |
| invariantUnder |
SL(2) action on binary quartic forms
ⓘ
change of homogeneous coordinates on the projective line ⓘ |
| mathematicalDomain |
computational invariant theory
ⓘ
representation theory of SL(2) ⓘ |
| namedAfter |
Alfred Clebsch
ⓘ
Siegfried Aronhold ⓘ |
| property |
generate the algebra of invariants of binary quartic forms over characteristic 0
ⓘ
polynomial functions in the coefficients of the form ⓘ remain unchanged under linear change of variables with determinant 1 ⓘ |
| relatedTo |
Hilbert basis of invariants
ⓘ
Hilbert’s work on invariants ⓘ absolute invariants of quartic forms ⓘ covariants of binary forms ⓘ discriminant of a binary quartic form ⓘ PSL(2,\mathbb{C}) ⓘ
surface form:
projective linear group PGL(2)
symbolic method in invariant theory ⓘ syzygies among invariants of binary forms ⓘ |
| usedFor |
classification of algebraic curves
ⓘ
classification of binary quartic forms ⓘ construction of moduli of binary quartics ⓘ describing orbits of binary quartics under SL(2) ⓘ distinguishing non-isomorphic quartic curves up to projective transformations ⓘ expressing projective invariants of four points on the projective line ⓘ study of projective equivalence classes of quartic forms ⓘ |
| usedIn |
classification of quartic polynomials up to linear fractional transformations
ⓘ
construction of moduli space of binary quartics ⓘ |
How these facts were elicited
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Input
Subject: Clebsch–Aronhold invariants Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.