##  [Bézout's Theorem](/bezouts-theorem-1) 

 Definition

A statement in algebraic geometry that two projective plane algebraic curves without a common component intersect in a number of points counted with multiplicity equal to the product of their degrees, over an algebraically closed field.

 

 

 

 

 

 





## Principle

Principle

Intersection number in the projective plane is governed by degrees: passing to projective closure and counting multiplicities (including intersections at infinity) yields degree(product) intersections when no component is shared.

 

 

 

 

 





## Demonstration

Demonstration

Two plane curves of degrees m and n (for example a conic of degree 2 and a cubic of degree 3) typically meet in m·n = 6 points in the projective plane when counted with multiplicity and over an algebraically closed field.

 

 

 

 

## Misapplication

Misapplication

Using the statement in the affine plane without accounting for points at infinity, ignoring multiplicity, or applying it over a non-algebraically-closed field without base change; also misusing it when the curves share a component.

 

 

 

 

 





## Consequence

Consequence

Gives a predictable count for intersections used in elimination theory and enumerative geometry, and leads to further refinements (intersection multiplicity, Bézout matrices, and conditions for common factors).

 

 

 

 

## Reversal

Reversal

If two curves intersect in fewer than m·n distinct points, that shortfall is explained by multiplicities, intersections at infinity, or a shared component; conversely, an excess signals multiplicities or degeneracies accounted for by the theorem.

 

 

 

 

 





## Boundary

Boundary

Assumes working in the projective plane over an algebraically closed field and that the curves have no common irreducible component; does not directly apply to higher-dimensional varieties without generalization or to schemes requiring refined intersection theory.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Often confused with Bézout's identity in number theory; within geometry it competes with local intersection concepts (e.g., scheme-theoretic multiplicity) that refine the naive point count.

 

 

 

 

 





## Synthesis

Synthesis

Bézout's Theorem organizes intersections of plane projective curves: after projective completion and counting multiplicities over an algebraically closed field, degrees multiply to give the total intersection count unless components are shared.