# Two semigroup rings associated to a finite set of germs of meromorphic functions

## TL;DR

This paper studies algebraic structures called semigroup rings derived from germs of meromorphic functions at a point, focusing on their properties and specific cases involving functions of finite order.

## Contribution

It introduces and analyzes two semigroup rings associated with germs of meromorphic functions, especially in the context of functions of finite order and their holomorphic properties.

## Key findings

- Characterization of the properties of the semigroup rings S^{hol} and ar S^{hol}
- Detailed analysis of the case where  is the field of meromorphic functions of order <1
- Results on the structure of these rings when functions have finitely many zeros and poles.

## Abstract

We fix $z_0\in\mathbb C$ and a field $\mathbb F$ with $\mathbb C\subset \mathbb F \subset \mathcal M_{z_0}:=$ the field of germs of meromorphic functions at $z_0$. We fix $f_1,\ldots,f_r\in \mathcal M_{z_0}$ and we consider the $\mathbb F$-algebras $S:=\mathbb F[f_1,\ldots,f_r]$ and $\overline S:=\mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}]$. We present the general properties of the semigroup rings \begin{align*} & S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\ & \overline S^{hol}:=\mathbb F[f^{\mathbf a}:=f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0], \end{align*} and we tackle in detail the case in which $\mathbb F=\mathcal M_{<1}$ is the field of meromorphic functions of order $<1$ and $f_j$'s are meromorphic functions over $\mathbb C$ of finite order with a finite number of zeros and poles.