Bezout Bound for Surfaces
Summary: §150 of the Appendix on Surfaces. If two surfaces have orders and , the projection of their intersection has order at most . Two planes: order-1 projection (a line). Plane + quadric: order-2 projection (a conic). Two quadrics: order-4 projection. The 3D lift of intersection-product-degree-bound (Book II Chapter 20 §§496–498), but in projection form — the spatial curve itself can have higher complexity, only its projection is bounded this simply.
Sources: appendix6, §150.
Last updated: 2026-05-12.
§150 — The statement
First of all, two first order surfaces, that is, two planes, have a projection of their intersection which is a first order line. We have also seen that the projection cannot have order higher than the second if one of the surfaces is first order and the other is second order. In a similar way it is clear that if one surface is third order and the other is first order, the projection cannot have order greater than the third, and so forth. However, two second order surfaces can have an intersection whose projections is fourth order or less. In general, if one surface has order and the other has order , then the projection of the intersection never has order greater than . (source: appendix6, §150)
So in projection:
| Order | Order | Max projection order |
|---|---|---|
| 1 | 1 | 1 |
| 1 | 2 | 2 |
| 2 | 2 | 4 |
| 1 | 3 | 3 |
| 2 | 3 | 6 |
| 3 | 3 | 9 |
Algebraic origin
The bound is a direct consequence of the elimination procedure (see intersection-of-two-surfaces §138). From the two equations of orders in , treating them as polynomials in (of degrees in respectively), the resultant has total degree in . This is the Sylvester resultant degree — same machinery as in indeterminate-multiplier-elimination.
Projection vs. intersection itself
A subtle point: the projection has order , but the space curve may have higher genus / arithmetic complexity. For two quadrics, the intersection is a space quartic curve of genus 1 (an elliptic curve in 3-space), whose projection to a generic plane is a degree-4 plane curve. So is the right number for the projected algebraic complexity, not for parametric complexity.
Tangency and lower-order projections
If the projection happens to factor, the curve consists of multiple lower-order pieces — this is the tangency / common-factor scenario of tangency-of-surfaces. For example, when two quadrics are tangent along a conic, the projection eliminant has a perfect-square factor, so the order drops below 4.
Cross-references
- Direct 3D analogue of intersection-product-degree-bound from Book II Chapter 20 §§496–498.
- The Bezout bound is the underlying combinatorial reason for the count in general-construction-by-parabola-and-conic §§499–504, where Euler engineered curve pairs to realize the bound exactly.
- A version of this bound (in projective space, counting multiplicities) is the modern Bezout’s theorem for surfaces: two algebraic surfaces of degrees with no common component meet in a curve of degree . Euler’s “projection” formulation is a weaker affine cousin.