Two semigroup rings associated to a finite set of meromorphic functions

Mircea Cimpoeaş 1
  • 1 Simion Stoilow Institute of Mathematics, Research unit 5, P.O.Box 1-764 Calea Griviţei street 21 014700, Bucharest, Romania
Mircea Cimpoeaş
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  • Simion Stoilow Institute of Mathematics, Research unit 5, P.O.Box 1-764 Calea Griviţei street 21 014700, Bucharest, Romania
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Abstract

We fix z0 ∈ ℂ and a field 𝔽 with ℂ ⊂ 𝔽 ⊂ 𝓜z0 := the field of germs of meromorphic functions at z0. We fix f1, …, fr ∈ 𝓜z0 and we consider the 𝔽-algebras S := 𝔽[f1, …, fr] and S¯:=F[f1±1,,fr±1]. We present the general properties of the semigroup rings

Shol:=F[fa:=f1a1frar:(a1,,ar)Nr and fa is holomorphic at z0],S¯hol:=F[fa:=f1a1frar:(a1,,ar)Zr and fa is holomorphic at z0],

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.

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Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process.

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