Bézout's Theorem: Counting Intersections of Plane Curves

The precise statement

Let k be an algebraically closed field and let C, D ⊂ ℙ²(k) be projective plane curves cut out by homogeneous polynomials F, G ∈ k[X, Y, Z] of degrees m and n respectively. Assume F and G have no common non-constant factor (equivalently, C and D share no irreducible component). Then C ∩ D is a finite set, and

∑_{P ∈ C ∩ D} I(P; C ∩ D) = m·n,

where I(P; C ∩ D) ∈ ℤ_≥1 is the intersection multiplicity of C and D at P, defined algebraically as I(P) = dim_k 𝒪_P/(F, G) — the k-dimension of the local ring of ℙ² at P modulo the ideal generated by the two forms (dehomogenized near P). All three ingredients are load-bearing: algebraically closed guarantees enough points exist, projective supplies the points at infinity, and multiplicity upgrades an inequality (≤ mn) into an exact equality (= mn).

The picture: why exactly mn

Think of a degree-m curve as the zero set of a polynomial that, restricted to a generic line, has m roots. Sweeping the second curve past the first, you expect the m 'branches' of C to each cross the n branches of D — giving mn crossings — like m horizontal strands woven through n vertical ones. Over ℝ, strands miss each other (no real solution), run off to infinity (parallel lines), or graze tangentially (a double crossing hidden as one). Each pathology subtracts visible points, which is why real affine geometry only ever sees ≤ mn.

The genius of the projective-complex setting is that it plugs every leak. Complex algebraic closure means strands can never simply 'miss': a polynomial always has its full quota of roots. Projective completion adds a line at infinity where parallel branches finally meet. And intersection multiplicity counts a tangency as the double point it really is. Close all three leaks and the mn crossings reappear, every single time.

Key idea of the proof: the resultant

The cleanest classical proof uses the resultant. Pick coordinates so that [0:1:0] lies on neither curve, and view F, G as polynomials in Y with coefficients in k[X, Z]. Their resultant R(X, Z) = Res_Y(F, G) is a homogeneous polynomial that vanishes at [X:Z] precisely when F(X,·,Z) and G(X,·,Z) share a root Y — i.e. exactly on the images of the intersection points under projection from [0:1:0]. A determinantal degree count (the Sylvester matrix is (m+n)×(m+n)) shows deg R = mn. Because k is algebraically closed, R factors into mn linear forms, and one proves each factor's multiplicity equals the sum of I(P) over points P above that direction.

The modern proof replaces resultants with the Hilbert polynomial of the graded ring S/(F,G), where S = k[X,Y,Z]. Since F, G form a regular sequence (no common factor), the exact Koszul resolution 0 → S(−m−n) → S(−m)⊕S(−n) → S → S/(F,G) → 0 forces the Hilbert polynomial of the quotient to be the constant mn — which is exactly the total intersection number by Serre's Tor formula.

Worked example: a line tangent to a parabola

Take the parabola C: Y = X² and the horizontal line D: Y = 0. Homogenize with Z: F = YZ − X² and G = Y. In the affine chart Z = 1, substitute Y = 0 into YZ − X² = 0 to get −X² = 0, so X = 0 with multiplicity 2. Thus the origin P = [0:0:1] is the only visible intersection, but I(P) = dim_k k[X,Y]_(0,0)/(Y − X², Y) = dim k[X]/(X²) = 2. Since m = deg F = 2 and n = deg G = 1, we need mn = 2 points — and the double point supplies exactly them. The line is tangent, so its two intersections have coalesced.

Now perturb: D': Y = ε (ε ≠ 0). Then X² = ε gives X = ±√ε, two distinct points, each simple. The double point split into two as we broke the tangency — vivid confirmation that multiplicity is exactly the number of points that merged, and that the total 2 is a deformation invariant.

Why the hypotheses matter

Drop 'algebraically closed'. Over ℝ, the circle X² + Y² = Z² and the line X = 2Z (both projective, no common component, m = 2, n = 1) should meet in mn = 2 points, but 4 + Y² = 1 has no real solution — the two intersection points are complex conjugates [2 : ±i√3 : 1]. Real geometry undercounts.

Drop 'projective'. Two distinct parallel lines in 𝔸²(ℂ) never meet, yet mn = 1. Their unique intersection sits at infinity on the line Z = 0; only the projective completion recovers it.

Drop 'no common component'. If F = G = a conic, then C ∩ D is the entire conic — infinitely many points — and the finite total mn is meaningless. This is why the resultant R vanishes identically exactly when F, G share a factor. Bézout's theorem is intimately tied to the regular-sequence condition and to intersection theory on surfaces, where it becomes deg C · deg D = (C · D) in the Chow ring / cohomology of ℙ².

Applications and significance

Bézout's theorem is the prototype of all of intersection theory, the backbone of enumerative algebraic geometry. It immediately yields the degree-genus formula's setup, bounds the number of singular points of a curve (a degree-d curve has at most (d−1)(d−2)/2 nodes), and underlies Plücker's formulas relating a curve's degree, class, and singularities. Chasles' count that a conic tangent to five given conics numbers 3264 is a Bézout-style intersection number in a compactified parameter space.

Beyond pure geometry: in elliptic-curve cryptography, the group law is defined by intersecting the cubic with lines — Bézout guarantees a third intersection point exists, making the addition well-defined. In numerical algebraic geometry, the Bézout number ∏dᵢ bounds the count of solutions to polynomial systems and seeds homotopy-continuation solvers. And the theorem generalizes: n hypersurfaces of degrees d₁,…,dₙ in ℙⁿ meet in ∏dᵢ points, the multiprojective and toric refinements (BKK bound via Newton polytopes) sharpening the count for structured systems.

How each hypothesis affects the count of intersection points for curves of degree m and n

Setting	What you can say	Example (line m=1 meets conic n=2)
Affine plane 𝔸²(ℝ), naive count	≤ mn, often fewer; no clean formula	Line x=3 meets circle x²+y²=1: 0 real points
Affine plane 𝔸²(ℂ)	≤ mn; points at infinity may be lost	Two parallel lines: 0 intersections (should be 1)
Projective plane ℙ²(ℂ), curves share a component	Infinitely many common points; theorem fails	C = D = a conic: whole curve is shared
Projective ℙ²(k̄), no common component, with multiplicity	Exactly mn (equality)	Tangent line to conic: 1 point of multiplicity 2 = 2
Projective ℙ²(k̄), counting distinct points only	≤ mn (Bézout inequality)	Line meeting conic transversally: 2 distinct points