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 (1)

Label	Occurrences
Bézout’s theorem canonical	1

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 ⓘ

Referenced by (1)

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

Noether’s theorem in algebraic geometry (Noether’s AF+BG theorem) → involves → Bézout’s theorem ⓘ

subject surface form: Noether’s AF+BG theorem