Bezout's Theorem

Learn the fundamental intersection theorem: two curves of degrees m and n meet in exactly m×n points (counted properly).

Bezout's Theorem: Counting Intersections

One of the most beautiful results in algebraic geometry: two algebraic curves of degrees m and n intersect in exactly m × n points — but only if we count correctly!

"Counting correctly" means: (1) working in projective space to catch points at infinity, (2) working over complex numbers to catch imaginary intersections, and (3) counting points with multiplicity when curves are tangent.

Demo 1: Counting Intersections

Select two curves and see their intersections. Bezout says: degree of C₁ times degree of C₂ equals the number of intersections.

Bezout's Theorem
1 × 2 = 2
Expected intersections (counted with multiplicity)
Line (y = x)
Circle
Visible intersections found:2

Demo 2: Intersection Multiplicity

When curves are tangent, the intersection point counts more than once. Drag the line to see how multiplicity changes.

Parabola: y = x²
Line: y = 1.00
Transverse intersection
2 points, each multiplicity 1
Bezout: 2 × 1 = 2 intersections (always!)
Counted: 1+1 = 2

Drag the green handle up and down. Watch how the line can be tangent to the parabola (multiplicity 2), cross it at two points (each multiplicity 1), or miss it entirely (two complex intersections).

Demo 3: Where's the Missing Intersection?

Sometimes you can't see all the intersections. Can you figure out where the "missing" ones are hiding?

Challenge: Find all 1 intersection!
deg 1×deg 1=1
Visible: 0 | Missing: 1

Bezout's Theorem Mastered!

You now understand one of algebraic geometry's foundational theorems:

  • The theorem: Curves of degrees m and n meet in m × n points
  • Multiplicity: Tangent intersections count multiple times
  • Points at infinity: Parallel lines meet at a projective point
  • Complex intersections: Some meet only over ℂ, not ℝ
  • Projective space: The natural setting where Bezout always works

Congratulations! You've completed the Algebraic Geometry introductory module. You've learned about algebraic curves, conic sections, and the powerful Bezout's Theorem that connects algebra and geometry.