Bézout's theorem

Remarks

One proof uses the resultant of the two polynomials with respect to one variable, after a change of coordinates. It is a homogeneous polynomial of degree $de$ in the other two variables, and its roots, with multiplicity, correspond to the intersection points. In intersection theory the proof is one line: $[C] = d h$ and $[D] = e h$ in $CH^\bullet(\mathbb{P}^2) = \Z[h]/(h^3)$, so $[C] \cdot [D] = de \, h^2$, and $h^2$ is the class of a point. Example: a line and a smooth conic meet in two points, or in one point of multiplicity $2$ when the line is tangent. Each hypothesis is needed. Over $\R$, the circle $x^2 + y^2 = z^2$ and the line $x = 2z$ have no common real point, but they meet in two complex points. In the affine plane, two parallel lines do not meet; in $\mathbb{P}^2$ they meet at a point at infinity. Without multiplicities, a tangent line meets a conic in only one point. If the polynomials have a common factor, the curves share a component and meet in infinitely many points. More generally, $n$ hypersurfaces of degrees $d_1, \ldots, d_n$ in $\mathbb{P}^n_k$ with finite intersection meet in $d_1 \cdots d_n$ points counted with multiplicity.

Going deeper

For a student

Bézout's theorem is the first result of intersection theory, and the model for the general principle that intersection numbers depend only on the classes of the cycles. It gives the bound of at most $de$ common points for any two plane curves without common component, over any field. Together with the genus formula it is the basic tool of the classical theory of plane curves.