Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field). Let $f,g\in k[x,y]$ be nonzero polynomials of degrees $m,n\in \mathbb N_0$, respectively, and suppose that $f$ and $g$ have no common irreducible factor in $k[x,y]$. Let $C=V(f)\subset \mathbb A_k^2$ and $D=V(g)\subset \mathbb A_k^2$. Let $F,G\in k[X,Y,Z]$ be the homogenizations of $f$ and $g$ to degrees $m$ and $n$, respectively, and let $\overline C=V_+(F)$ and $\overline D=V_+(G)$ be their projective closures in $\mathbb P_k^2$. Then

\begin{align*} \sum_{p\in C(k)\cap D(k)} I_p(C,D)+\sum_{p\in \overline C(k)\cap\overline D(k)\cap V_+(Z)} I_p(\overline C,\overline D)=mn, \end{align*}

where the first sum uses the affine chart $Z\ne 0$ identified with $\mathbb A_k^2$, and $I_p$ denotes the local intersection multiplicity of the two plane curves at $p$ in the corresponding affine or projective local ring.