Editing Bézout's theorem (section)

==Multiplicity==
{{main|Multiplicity (mathematics)|Intersection theory|Intersection number}}
The concept of multiplicity is fundamental for Bézout's theorem, as it allows having an equality instead of a much weaker inequality.

Intuitively, the multiplicity of a common zero of several polynomials is the number of zeros into which the common zero can split when the coefficients are slightly changed. For example, a tangent to a curve is a line that cuts the curve at a point that splits in several points if the line is slightly moved. This number is two in general (ordinary points), but may be higher (three for [[inflection point]]s, four for [[undulation point]]s, etc.). This number is the "multiplicity of contact" of the tangent.

This definition of a multiplicities by deformation was sufficient until the end of the 19th century, but has several problems that led to more convenient modern definitions: Deformations are difficult to manipulate; for example, in the case of a [[zero of a function|root]] of a [[univariate polynomial]], for proving that the multiplicity obtained by deformation equals the multiplicity of the corresponding linear factor of the polynomial, one has to know that the roots are [[continuous function]]s of the coefficients. Deformations cannot be used over [[field (mathematics)|fields]] of [[positive characteristic]]. Moreover, there are cases where a convenient deformation is difficult to define (as in the case of more than two plane curves having a common intersection point), and even cases where no deformation is possible.{{citation needed|date=June 2020}}

Currently, following [[Jean-Pierre Serre]], a multiplicity is generally defined as the [[length of a module|length]] of a [[local ring]] associated with the point where the multiplicity is considered.{{sfn|Serre|1965}} Most specific definitions can be shown to be special case of Serre's definition.

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.