---
tag: TT7R63MV96
kind: definition
name: Meromorphic function of several variables
msc: [32A20]
defines: W6FTSF57T1
uses: [507C34W0ZT, 02BH3MCBQB]
sources:
  - url: https://en.wikipedia.org/wiki/Meromorphic_function
---

## Definition

Let $D \subseteq \C^n$ be an [[open set|02BH3MCBQB]].
A *meromorphic function* on $D$ is given by an open cover $(U_i)_{i \in I}$ of $D$ by connected open sets.
For each $i$ it gives two [[holomorphic functions|507C34W0ZT]] $g_i, h_i$ on $U_i$ such that:

1. $h_i$ is not identically zero on $U_i$;
2. $g_i h_j = g_j h_i$ on $U_i \cap U_j$ for all $i, j \in I$.

Two such data $(U_i, g_i, h_i)_{i \in I}$ and $(V_k, p_k, q_k)_{k \in K}$ define the same meromorphic function when $g_i q_k = p_k h_i$ on $U_i \cap V_k$ for all $i$ and $k$.

## Remarks

On the dense open set where some $h_i$ is nonzero, a meromorphic function $f$ is the holomorphic function $g_i / h_i$.
Its polar set and its set of indeterminacy points are analytic subsets; for $n \ge 2$ the indeterminacy set has codimension at least $2$.

Example: on $D = \C^2$ take the single set $U_1 = \C^2$ with $g_1(z) = z_1$ and $h_1(z) = z_2$.
Clause 1 holds since $z_2$ is not the zero function, and clause 2 is empty, so $z_1 / z_2$ is meromorphic on $\C^2$.

Non-example: $e^{1/z_1}$, defined where $z_1 \ne 0$, is not the restriction of a meromorphic function on $\C^2$.
Suppose it were $g/h$ near a point $(0, c)$.
Then, for a suitable fixed $z_2$ close to $c$, the one-variable function $z_1 \mapsto e^{1/z_1}$ would be meromorphic at $0$.
But it has an essential singularity there.

For $n = 1$ this definition gives the usual [[meromorphic functions|0ZZ7ZYDR8W]] on an open set of $\C$.
In the language of sheaves, a meromorphic function is a global section of the sheaf of quotient fields of the sheaf of holomorphic functions.