arXiv:1907.12099v1 [math.CV] 28 Jul 2019 Two semigroup rings associated to a finite set of germs of meromorphic functions Mircea Cimpoeaş July 30, 2019 Abstract We fix z 0 ∈ C and a field F with C ⊂ F ⊂M z 0 := the field of germs of mero- morphic functions at z 0 . We fix f 1 ,...,f r ∈M z 0 and we consider the F-algebras S := F[f 1 ,...,f r ] and S := F[f ±1 1 ,...,f ±1 r ]. We present the general properties of the semigroup rings S hol := F[f a := f a 1 1 ··· f ar r :(a 1 ,...,a r ) ∈ N r and f a is holomorphic at z 0 ], S hol := F[f a := f a 1 1 ··· f ar r :(a 1 ,...,a r ) ∈ Z r and f a is holomorphic at z 0 ], and we tackle in detail the case in which F = M <1 is the field of meromorphic functions of order < 1 and f j ’s are meromorphic functions over C of finite order with a finite number of zeros and poles. 2010 MSC: 30D30; 30D20; 16S36. Keywords: meromorphic functions, entire functions, semigroup rings. 1 Introduction Let z 0 ∈ C and let g be a holomorphic function at z 0 , that is g is holomorphic on an open domain U ⊂ C with z 0 ∈ U . Replacing g (z) with g (z − z 0 ), we can assume that z 0 =0. Given two holomorphic functions g 1 and g 2 at 0 we say that g 1 ∼ g 2 if there exist an open domain U ∋ z 0 such that g 1 | U = g 2 | U . ∼ is an equivalence relation. A class of equivalence of ∼ is called a germ of holomorphic function. We denote O 0 the ring of germs of holomorphic functions at 0. It is well known that O 0 ∼ = C{z} = { +∞  n=0 a n z n : 1 lim sup n n  |a n | > 0}, the ring of convergent power series, which is an one dimensional local regular ring with the maximal ideal m = zC{z}. 1