Let $k$ be an [algebraically closed field](/page/Algebraically%20Closed%20Field), let $F,G\in k[x,y]$ be nonzero polynomials, and let $p\in \mathbb A_k^2(k)$ satisfy $F(p)=G(p)=0$. Let $C:=V(F)$ and $D:=V(G)$ be the corresponding affine plane curves. Assume that $C$ and $D$ are nonsingular at $p$, and that their tangent lines at $p$ are distinct. If the local intersection multiplicity is defined by

\begin{align*} I_p(F,G):=\dim_k \mathcal O_{\mathbb A_k^2,p}/(F,G), \end{align*}

then

\begin{align*} I_p(F,G)=1. \end{align*}