Bezout's theorem
/ bay-ZOO /
How many times can two curves cross? Two lines meet once. A line and a circle meet twice (or seem to miss, or just graze). Two ellipses can meet in up to four points. There is a beautiful pattern hiding here, and Bezout's theorem states it exactly: two algebraic curves of degrees m and n meet in exactly m times n points — no more, and, properly counted, no fewer. Line (degree 1) and conic (degree 2): 1 times 2 = 2 points. Two conics: 2 times 2 = 4 points. The intersection count is just the product of the degrees.
The catch is in those three honest words 'properly counted', because the clean statement only holds once you fix three real-world annoyances. First, work over the COMPLEX numbers, not just the reals: a line that visibly misses a circle still meets it in two complex points. Second, work in the PROJECTIVE plane, adding points at infinity: two parallel lines, which 'never meet', do meet — at a single point at infinity, restoring 1 times 1 = 1. Third, count intersections with MULTIPLICITY: when a line is tangent to a circle it touches at one visible point but that point counts TWICE, because the two intersection points have merged. With all three conventions in place, the count is exactly m times n, with no exceptions whatsoever.
Bezout's theorem is the prototype of how algebraic geometry tames intersection by turning it into arithmetic of degrees, and it generalizes to surfaces and higher dimensions. It also explains familiar facts at a stroke: a quadratic (degree 2) meets the x-axis (degree 1) in 2 points, which is exactly why a quadratic has 2 roots over the complex numbers, counted with multiplicity. The common misreading is to apply the simple product over the reals and feel cheated when a line 'misses' a circle. Nothing is wrong — the missing two points are complex, and the theorem was never a promise about visible real crossings.
Take the circle x^2 + y^2 = 1 and the line y = 1, which touches it at the top. Substitute: x^2 + 1 = 1, so x^2 = 0. There is one visible point, (0, 1), but the equation x^2 = 0 has a DOUBLE root at x = 0, so it counts as 2 intersections. Degree 2 times degree 1 = 2, exactly as Bezout promised — the tangency hides the second point inside a multiplicity.
A tangency is two intersection points that have merged; multiplicity keeps the count honest.
The product m times n is the MAXIMUM and the exact count only with the three conventions (complex, projective, with multiplicity). Drop any one and the number can fall short over the reals — that is not a failure of the theorem but a reminder of its hypotheses.