Bézout's theorem

In algebraic geometry, the general statement of Bézout's theorem applies to the points of intersection of plane curves X of degree m and Y of degree n. It asserts that the number of intersections, counted by intersection multiplicity, is precisely mn, except in case X and Y have a common component. Therefore mn is the maximum finite number of intersection points. Here degree of a curve C means the degree of the polynomial defining it.

The special case where one of the curves is a line is a version of the fundamental theorem of algebra. For example, the parabola defined by y - x² = 0 has degree 2; the line y - 2x = 0 has degree 1, and they meet in exactly two points.

From the case of lines, with m and n both 1, it is clear that one must work in the projective plane; to allow for higher degree cases one is forced to set the theorem in P²_K over an algebraically closed field K. Only under those conditions can this be expected to be a valid theorem. In the real projective plane we can for example have two nested circles that don't intersect at all. The representative case is two ellipses cutting in four points.

One must count intersections with correct multiplicities, including tangencies. A tangent line to a circle should count as two intersections.

See also:

• Bézout's lemma[?]

External links

• http://www2.math.uic.edu/~jan/srvart/node7