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?
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Sign up to join this communityI 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?
If we work in the category of holomorphic functions, then we can give the following definition, that generalises to the complex-analytic setting the classical intersection multiplicity used in algebraic geometry.
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 $\boldsymbol{z_0}.$
See for instance
T. De Jong, G. Pfister: Local Analytic Geometry: Basic Theory and Applications, Chapter 5.