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

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 $f_1,\dots ,f_r\in {\mathcal{M}}_{z_0}$ and we consider the $\mathbb{F}$-algebras $S:=\mathbb{F}[f_1,\dots ,f_r]$ and $\overline{S}:=\mathbb{F}[f_1^{\pm 1},\dots ,f_r^{\pm 1}]$. 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.

arXiv comment11 pages

References in corpus (1)

• On the semigroup ring of holomorphic Artin L-functions