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

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

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

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

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

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

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

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

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.