On the semigroup ring of holomorphic Artin L-functions

TL;DR

This paper explores the algebraic structure of semigroup rings formed by Artin L-functions, characterizing their properties as toric rings and analyzing cases with simple zeros and poles at specific points.

Contribution

It provides a detailed description of the semigroup ring structure of Artin L-functions and characterizes the toric ideal in cases with simple zeros and poles.

Findings

Semigroup ring is isomorphic to a toric ring in the given setting.

Characterization of the semigroup ring when the toric ideal is zero.

Description of the ring and ideal when functions have only simple zeros and poles.

Abstract

Let $K/\mathbb Q$ be a finite Galois extension and let $\chi_1,\ldots,\chi_r$ be the irreducible characters of the Galois group $G:=Gal(K/\mathbb Q)$. Let $f_1:=L(s,\chi_1),\ldots,f_r:=L(s,\chi_r)$ be their associated Artin L-functions. For $s_0\in \mathbb C\setminus\{1\}$, we denote $Hol(s_0)$ the semigroup of Artin $L$-functions, holomorphic at $s_0$. Let $\mathbb F$ be a field with $\mathbb C \subseteq \mathbb F \subseteq \mathcal M_{<1}:=$ the field of meromorphic functions of order $<1$. We note that the semigroup ring $\mathbb F[Hol(s_0)]$ is isomorphic to a toric ring $\mathbb F[H(s_0)]\subseteq \mathbb F[x_1,\ldots,x_r]$, where $H(s_0)$ is an affine subsemigroup of $\mathbb N^r$ minimally generated by at least $r$ elements, and we describe $\mathbb F[H(s_0)]$ when the toric ideal $I_{H(s_0)}=(0)$. Also, we describe $\mathbb F[H(s_0)]$ and $I_{H(s_0)}$ when $f_1,\ldots,f_r$ have only simple zeros and simple poles at $s_0$.