Bézout's theorem
/ bay-ZOO /
A line meets a parabola in two points, a line meets a cubic in three, two conics meet in four. There is a beautifully simple pattern: the number of intersection points of two plane curves equals the product of their degrees. Bézout's theorem makes this exact — but only after you do three honest things: work in the projective plane (to catch intersections at infinity), use an algebraically closed field (so the points actually exist), and count each intersection with its multiplicity (so tangencies are weighted correctly).
Precisely: let C and D be plane projective curves over an algebraically closed field, of degrees m and n, with no common component. Then the total number of intersection points, each counted with its intersection multiplicity, is exactly mn: the sum over points p of I_p(C, D) equals mn. All three corrections are essential and each repairs a way the naive count fails. Working in P^2 rather than A^2 supplies the points at infinity (two parallel lines, degree 1 each, meet at one point at infinity, recovering 1 x 1 = 1). Algebraic closure guarantees the roots exist (a line and a circle that 'miss' over R meet at complex points). And multiplicity makes a tangency between a line and a conic count as 2, not 1, so the line-conic total is the correct 1 x 2 = 2.
Bézout is the prototype of enumerative geometry — counting geometric configurations by degree — and the engine behind facts like 'a smooth cubic plane curve has nine inflection points' and the projective duality of curves. It generalizes: n hypersurfaces in P^n of degrees d_1, ..., d_n meeting properly intersect in d_1 x ... x d_n points counted with multiplicity, the higher-dimensional Bézout. Honest caveats. The 'no common component' hypothesis is not optional — two curves sharing a component intersect in infinitely many points and the theorem says nothing. The equality is exact only with all three corrections in place; drop projectivity, closure, or multiplicity and you get an inequality at best, and the clean number mn dissolves.
A line (degree 1) and a smooth conic (degree 2) should meet in 1 x 2 = 2 points. A secant line cuts the conic at two distinct points; a tangent line touches at one point but with multiplicity 2; and a line that misses the conic over R still meets it in two complex points. In every case the count is exactly 2 in P^2 over C.
Projective + algebraically closed + counted with multiplicity makes the intersection number exactly the product of degrees.
The equality mn requires all three of: projective plane, algebraically closed field, and multiplicity, plus no shared component. Curves with a common component meet in infinitely many points and the theorem does not apply.