# Intersection multiplicity in the non-algebraic case

I know the definition of intersection multiplicity in algebraic geometry. However, I think it is possible to define it for some sort of non-algebraic functions such as $y=\sin x$.

How to define this?

Definition. Let $f$, $g$ be two holomorphic functions defined in a neighborhood of a point $\boldsymbol{z_0} \in \mathbb{C}^2$ and let $$X \colon \{f =0\}, \quad Y \colon \{g=0\}$$ be the corresponding (germs of) analytic varieties. Then the intersection multiplicity of $X$ and $Y$ at the point $\boldsymbol{z_0}$ is $$m(X, \, Y; \, \boldsymbol{z_0})= \dim_{\mathbb{C}} \mathcal{O}_{\mathbb{C}^2, \,\boldsymbol{z_0} }/(f, \, g),$$ where $\mathcal{O}_{\mathbb{C}^2, \,\boldsymbol{z_0} }$ stands for the analytic local ring of $\mathbb{C}^2$ at $\boldsymbol{z_0}.$
• It is worth pointing out that this definition of local multiplicity generalizes directly to the case of an arbitrary smooth map $f:\mathbb{R}^n\to\mathbb{R}^m$ and any point $p\in\mathbb{R}^n$, it's just that the multiplicity need not be finite. Just let $A_p$ be the ring of gerns of smooth functions defined in a neighborhood of $p$ and define the local multiplicity of $f$ at $p$ to be the dimension of the vector space $$A_p/(f^1,\ldots,f^m).$$ For properties of this multiplicity, see any treatment of the Malgrange Preparation Theorem. Jan 8 '17 at 15:07