We have hinted several times at a multiplication theorem for two intersecting algebraic curves, and it is now time to make a promise. In this chapter, we state such a theorem and sketch its proof.

Looking at the parabola y = x2 and the line y = 1 suggests what is needed to accomplish our aim. These curves have degree two and one, and intersect in 2 · 1 points. As we parallel-translate the line downward, the two points of intersection approach each other, and when the line coincides with the x-axis, the points have coalesced, “piling up on each other” at the origin. It is natural to count both intersection points, counting the origin with multiplicity two. When the line is pushed further to y = −ϵ(ϵ > 0), the curves' intersection points are found from the solutions to y = x2 and y = −ϵ,and these are x = ±i √ϵ. The two intersection points are therefore (+i√ϵ, −ϵ) and (−i√ϵ, −ϵ. In the overall downward sweep, the two intersection points begin as real and distinct, approach each other until they meet, then continue as imaginary and distinct. This suggests that working over ℂ instead of ℝ allows us to see and keep track of the intersections.