In the case of two variables and in the case of affine hypersurfaces, if multiplicities and points at infinity are not counted, this theorem provides only an upper bound of the number of points, which is almost always reached. This bound is often referred to as the Bézout bound.
There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language or algebraic geometry. Three algebraic proofs are sketched below.
Bézout's theorem has been generalized as the so-called multi-homogeneous Bézout theorem.
Two conic sections generally intersect in four points, some of which may coincide. To properly account for all intersection points, it may be necessary to allow complex coordinates and include the points on the infinite line in the projective plane. For example:
The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality.
In the case of Bézout's theorem, the general intersection theory can be avoided, as there are proofs (see below) that associate to each input data for the theorem a polynomial in the coefficients of the equations, which factorizes into linear factors, each corresponding to a single intersection point. So, the multiplicity of an intersection point is the multiplicity of the corresponding factor. The proof that this multiplicity equals the one that is obtained by deformation, results then from the fact that the intersection points and the factored polynomial depend continuously on the roots.
Proving the equality with other definitions of intersection multiplicities relies on the technicalities of these definitions and is therefore outside the scope of this article.
This proof of Bézout's theorem seems the oldest proof that satisfies the modern criteria of rigor.
Bézout's theorem can be proved by recurrence on the number of polynomials by using the following theorem.
Beside allowing a conceptually simple proof of Bézout's theorem, this theorem is fundamental for intersection theory, since this theory is essentially devoted to the study of intersection multiplicities when the hypotheses of the above theorem do not apply.