Editing Bézout's theorem (section)

==Statement==
===Plane curves===
Suppose that ''X'' and ''Y'' are two plane [[projective plane|projective]] curves defined over a [[field (mathematics)|field]] ''F'' that do not have a common component (this condition means that  ''X'' and ''Y'' are defined by polynomials, without [[polynomial greatest common divisor|common divisor]] of positive degree). Then the total number of intersection points of ''X'' and ''Y'' with coordinates in an [[algebraically closed field]] ''E'' that contains ''F'', counted with their [[intersection number|multiplicities]], is equal to the product of the degrees of ''X'' and ''Y''.

===General case===
The generalization in higher dimension may be stated as:

Let ''n'' [[projective hypersurface]]s be given in a [[projective space]] of dimension ''n'' over an algebraically closed field, which are defined by ''n'' [[homogeneous polynomial]]s in ''n'' + 1 variables, of degrees <math>d_1, \ldots,d_n.</math> Then either the number of intersection points is infinite, or the number of intersection points, counted with multiplicity, is equal to the product <math>d_1 \cdots d_n.</math> If the hypersurfaces are  in relative [[general position]], then there are <math>d_1 \cdots d_n</math> intersection points, all with multiplicity 1.

There are various proofs of this theorem, which either are expressed in purely algebraic terms, or use the language of [[algebraic geometry]]. Three algebraic proofs are sketched below.

Bézout's theorem has been generalized as the so-called [[multi-homogeneous Bézout theorem]].

===Affine case===
The affine case of the theorem is the following statement, that was proven in 1983 by [[David Masser]] and [[Gisbert Wüstholz]].{{sfn| Masser|Wüstholz|1983}}

''Consider {{mvar|n}} [[affine variety|affine hypersurface]]s that are defined  over an algebraically closed field by {{mvar|n}} [[ polynomial]]s in {{mvar|n}}  variables, of degrees <math>d_1, \ldots,d_n.</math> Then either the number of intersection points is infinite, or the number of intersection points, counted with their multiplicities, is at most the product <math>d_1 \cdots d_n.</math>'' If the hypersurfaces are in relative [[general position]], then there are exactly <math>d_1 \cdots d_n</math> intersection points, all with multiplicity 1.

The last assertion is a corollary of Bézout's theorem, but the first assertion is not, because of the possibility of a finite number of intersection points in the affine space, together with infinitely many intersection points at infinity.

This theorem a corollary, not explicitly stated, of a more general statement proved by Masser and Wüstholz. 

For stating the general result, one has to recall that the intersection points form an [[algebraic set]], and that there is a finite number of intersection points if and only if all component of the intersection have a zero [[dimension of an algebraic variety|dimension]] (an algebraic set of positive dimension has an infinity of points over an algebraically closed field). An intersection point is said ''isolated'' if it does not belong to a component of positive dimension of the intersection; the terminology make sense, since an isolated intersection point has neighborhoods (for [[Zariski topology]] or for the usual topology in the case of complex hypersurfaces) that does not contain any other intersection point.

Consider {{mvar|n}} projective hypersurfaces that are defined  over an algebraically closed field by {{mvar|n}} [[homogeneous polynomial]]s in <math>n+1</math>  variables, of degrees <math>d_1, \ldots,d_n.</math> Then, the sum of the multiplicities of their isolated intersection points   is at most the product <math>d_1 \cdots d_n.</math> The result remains valid for any number {{mvar|m}} of hypersurfaces, if one sets <math>d_{m+1}=0</math> in the case <math>m<n,</math> and, otherwise, if one orders the degrees for having <math>d_2\ge d_3\ge\cdots \ge d_m \ge d_1.</math> That is, there is no isolated intersection point if <math>m<n,</math> and, otherwise, the bound is the product of the smallest degree and the <math>n-1</math> largest degrees.