3
$\begingroup$

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?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
7
$\begingroup$

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.

Improve this answer
$\endgroup$
2
  • 2
    $\begingroup$ 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. $\endgroup$
    – Robert Bryant
    Commented Jan 8, 2017 at 15:07
  • 1
    $\begingroup$ Good point, thanks. In the real case, Malgrange's theorem plays the role of Weierstrass PreparationTheorem. By the way, it is surprising (at least for a non-expert like me) that we have such a result in the smooth case, not only in the real analytic one. $\endgroup$
    – Francesco Polizzi
    Commented Jan 8, 2017 at 15:14

You must log in to answer this question.