meromorphic functions of several variables

Definition.

Let $\mathrm{\Omega }\subset {ℂ}^n$ be a domain and let $h:\mathrm{\Omega }\to ℂ$ be a function. $h$ is called if for each $p\in \mathrm{\Omega }$ there exists a neighbourhood $U\subset \mathrm{\Omega }$ ($p\in U$) and two holomorphic (http://planetmath.org/HolomorphicFunctionsOfSeveralVariables) functions $f,g$ defined in $U$ where $g$ is not identically zero, such that $h=f/g$ outside the set where $g=0$.

Note that $h$ is really defined only outside of a complex analytic subvariety. Unlike in one variable, we cannot simply define $h$ to be equal to $\mathrm{\infty }$ at the poles and expect $h$ to be a continuous mapping to some larger space (the Riemann sphere in the case of one variable). The simplest counterexample in ${ℂ}^2$ is $(z,w)\mapsto z/w$, which does not have a unique limit at the origin. The set of points where there is no unique limit, is called the indeterminancy set. That is, the set of points where if $h=f/g$, and $f$ and $g$ have no common factors, then the indeterminancy set of $h$ is the set where $f=g=0$.

Title	meromorphic functions of several variables
Canonical name	MeromorphicFunctionsOfSeveralVariables
Date of creation	2013-03-22 16:01:10
Last modified on	2013-03-22 16:01:10
Owner	jirka (4157)
Last modified by	jirka (4157)
Numerical id	4
Author	jirka (4157)
Entry type	Definition
Classification	msc 32A20
Defines	indeterminancy set