In algebraic geometry, the 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 - x^{2} = 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 P^{2}_{K} over an algebraically closed field K.
Any conic should meet the line at infinity at two points according to the theorem. A hyperbola meets it at two real points corresponding to the two directions of the asymptotes. An ellipse meets it at two complex points which are conjugate to one another---in the case of a circle, the points (1:i:0) and (1:-i:0). A parabola meets it at only one point, but it is a point of tangency and therefore counts twice.
In general, two conics meet in four points. The following pictures show examples in which the circle x^{2}+y^{2}-1=0 meets another ellipse in fewer intersection points because at least one of them has multiplicity greater than 1:
See also: