Two semigroup rings associated to a finite set of meromorphic functions

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.