Bézout’s theorem

E705365

theorem in algebraic geometry

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.

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All labels observed (3)

Label	Occurrences
Bézout’s theorem canonical	1
Bézout’s theorem for projective space of higher dimension	1
Bézout’s theorem for systems of homogeneous polynomials	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.
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
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 ⓘ linked to: Chow groups 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 ⓘ linked to: Bézout’s theorem Bézout’s theorem for systems of homogeneous polynomials ⓘ linked to: Bézout’s theorem
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 ⓘ
relatedTo	Bézout matrix ⓘ linked to: Bezout matrix Hilbert’s Nullstellensatz ⓘ 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

Referenced by (3)

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

Noether’s AF+BG theorem → involves → Bézout’s theorem ⓘ

subject linked to: Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem)

Bézout’s theorem → hasVariant → Bézout’s theorem for projective space of higher dimension ⓘ

linked to: Bézout’s theorem

Bézout’s theorem → hasVariant → Bézout’s theorem for systems of homogeneous polynomials ⓘ