# Bézout's theorem

In algebraic geometry , the classical **Bézout theorem describes** the number of points of intersection of plane algebraic curves . It was formulated and proven by Étienne Bézout in the 18th century (within the framework of the more lax claims of the time).

## statement

Let be an algebraically closed body and let and two projective plane curves in two-dimensional projective space without common components. Then:

where denotes the number of cuts .

## Inferences

- Two projective plane curves and always intersect in at least one point and at most in different points.
- The inequality holds for affine plane curves and without common components .

## generalization

A generalization for algebraic varieties is as follows:

Be , algebraic varieties of degree or in -dimensional projective space . Furthermore, be a variety of dimension .

Then is .

## Web links

**Wikiversity: A proof of the theorem in the even case**- course materials

## literature

Klaus Hulek: *Elementare Algebraische Geometrie* , 1st edition, 2000, ISBN 978-3-528-03156-5 , pp. 145-146.