Bézout's theorem

In the algebraic geometry of the classical set of Bézout describes the number of points of intersection of plane algebraic curves. It was formulated by Étienne Bézout in the 18th century and proved ( under the laxer claims that time ).

Statement

Be an algebraically closed field and let and two projective plane curves in two-dimensional projective space with no common components. Then:

The average number referred.

Conclusions

• Two projective plane curves and intersecting always at least one point and a maximum number of points.
• For affine plane curves without common components, the inequality holds.

Generalization

A generalization of algebraic varieties is as follows:

Be algebraic varieties of degree or in -dimensional projective space. It should also be a variety of dimension.

Then is.