Bézout’s theorem
E705365
Bézout’s theorem is a fundamental result in algebraic geometry stating that, over an algebraically closed field, the number of intersection points of two projective plane curves (counted with multiplicity) equals the product of their degrees.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Bézout’s theorem canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T7997802 — 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: Bézout’s theorem Context triple: [Noether’s AF+BG theorem, involves, Bézout’s theorem]
-
A.
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.
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B.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
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C.
Bombieri–Pila determinant method
The Bombieri–Pila determinant method is a technique in analytic and Diophantine geometry used to obtain upper bounds on the number of rational or integral points of bounded height lying on algebraic curves or more general sets.
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D.
Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
-
E.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
- 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: Bézout’s theorem Target entity description: Bézout’s theorem is a fundamental result in algebraic geometry stating that, over an algebraically closed field, the number of intersection points of two projective plane curves (counted with multiplicity) equals the product of their degrees.
-
A.
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.
-
B.
Plücker formulas
Plücker formulas are classical algebraic geometry relations that connect the degree and singularities of plane algebraic curves with the invariants of their dual curves.
-
C.
Bombieri–Pila determinant method
The Bombieri–Pila determinant method is a technique in analytic and Diophantine geometry used to obtain upper bounds on the number of rational or integral points of bounded height lying on algebraic curves or more general sets.
-
D.
Chevalley’s theorem in algebraic geometry
Chevalley’s theorem in algebraic geometry is a fundamental result stating that the image of a morphism of finite type between schemes (or varieties) is a constructible set, playing a key role in understanding how geometric properties behave under mappings.
-
E.
Chevalley–Warning theorem
The Chevalley–Warning theorem is a result in number theory and algebraic geometry that gives conditions under which systems of polynomial equations over finite fields must have nontrivial solutions.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf | theorem in algebraic geometry ⓘ |
| appliesOver | algebraically closed field ⓘ |
| appliesTo |
irreducible projective plane curves
ⓘ
projective plane curves ⓘ reducible projective plane curves without common components ⓘ |
| assumes |
curves are given by homogeneous polynomials
ⓘ
no common component between the two curves ⓘ |
| conclusion | sum of intersection multiplicities equals product of degrees ⓘ |
| context |
Chow ring of projective space
NERFINISHED
ⓘ
homogeneous coordinates ⓘ projective completion of affine curves ⓘ |
| countsIntersections | with multiplicity ⓘ |
| failsIf | the two curves share a common component ⓘ |
| field | algebraic geometry ⓘ |
| generalizes | intersection counting for lines and conics in the projective plane ⓘ |
| hasFormulation |
in terms of degrees of divisors on projective curves
ⓘ
in terms of intersection numbers in the Chow ring ⓘ |
| hasVariant |
Bézout’s theorem for projective space of higher dimension
NERFINISHED
ⓘ
Bézout’s theorem for systems of homogeneous polynomials NERFINISHED ⓘ |
| historicalPeriod | 18th century mathematics ⓘ |
| implies | two projective plane curves of degrees m and n intersect in mn points counting multiplicities ⓘ |
| influenced | development of modern intersection theory ⓘ |
| namedAfter | Étienne Bézout NERFINISHED ⓘ |
| relatedTo |
Bézout matrix
NERFINISHED
ⓘ
Hilbert’s Nullstellensatz NERFINISHED ⓘ intersection theory ⓘ resultants of polynomials ⓘ |
| relatesConcept |
algebraically closed fields
ⓘ
degree of a projective plane curve ⓘ intersection multiplicity ⓘ projective space ⓘ |
| requires |
curves to be considered in the projective plane
ⓘ
curves to be defined over an algebraically closed field ⓘ |
| requiresMultiplicity |
intersection points at infinity to be counted
ⓘ
tangent intersections to be counted with multiplicity greater than one ⓘ |
| specialCase |
a line and a conic intersect in two points counting multiplicities
ⓘ
two conics intersect in four points counting multiplicities ⓘ two distinct projective lines intersect in exactly one point ⓘ |
| statesThat | the number of intersection points of two projective plane curves equals the product of their degrees ⓘ |
| typeOfResult | global intersection formula ⓘ |
| usedFor |
analyzing singular intersections of curves
ⓘ
bounding the number of solutions of polynomial equations ⓘ counting complex solutions to systems of two bivariate polynomials ⓘ |
| usedIn |
computational algebraic geometry
ⓘ
enumerative geometry ⓘ proofs of existence of intersection points of curves ⓘ theory of polynomial systems ⓘ |
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: Bézout’s theorem Description of subject: Bézout’s theorem is a fundamental result in algebraic geometry stating that, over an algebraically closed field, the number of intersection points of two projective plane curves (counted with multiplicity) equals the product of their degrees.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Noether’s AF+BG theorem