Affiliation:
1. Simion Stoilow Institute of Mathematics , Research unit 5, P.O.Box 1-764 Calea Griviţei street 21 014700 , Bucharest , Romania
Abstract
Abstract
We fix z
0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜
z
0
:= the field of germs of meromorphic functions at z
0. We fix f
1, …, fr
∈ 𝓜
z
0
and we consider the 𝔽-algebras S := 𝔽[f
1, …, fr
] and
S
¯
:=
F
[
f
1
±
1
,
…
,
f
r
±
1
]
.
$\begin{array}{}
\overline S: = \mathbb F[f_1^{\pm 1},\ldots,f_r^{\pm 1}].
\end{array} $
We present the general properties of the semigroup rings
S
h
o
l
:=
F
[
f
a
:=
f
1
a
1
⋯
f
r
a
r
:
(
a
1
,
…
,
a
r
)
∈
N
r
and
f
a
is holomorphic at
z
0
]
,
S
¯
h
o
l
:=
F
[
f
a
:=
f
1
a
1
⋯
f
r
a
r
:
(
a
1
,
…
,
a
r
)
∈
Z
r
and
f
a
is holomorphic at
z
0
]
,
$$\begin{array}{}
\displaystyle S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb N^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],\\\overline S^{hol}: = \mathbb F[f^{\mathbf a}: = f_1^{a_1}\cdots f_r^{a_r}: (a_1,\ldots,a_r)\in\mathbb Z^r \text{ and }f^{\mathbf a}\text{ is holomorphic at }z_0],
\end{array} $$
and we tackle in detail the case 𝔽 = 𝓜<1, the field of meromorphic functions of order < 1, and fj
’s are meromorphic functions over ℂ of finite order with a finite number of zeros and poles.