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Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/w980545678091g Pluriharmonic Definable Functions in Some oMinimal Expansions of the Real Field
Berraho, M.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4.
Abstract:
In this paper, we first try to solve the following problem: If a pluriharmonic function \(f\) is definable in an arbitrary ominimal expansion of the structure of the real field \(\overline{\mathbb{R}}:=(\mathbb{R},+,,.,0,1,<)\), is this function locally the real part of a holomorphic function which is definable in the same expansion? In Proposition 2.1 below, we prove that this problem has a positive answer if the Weierstrass division theorem holds true for the system of the rings of real analytic definable germs at the origin of \(\mathbb{R}^n\). We obtain the same answer for an ominimal expansion of the real field which is pfaffian closed (Proposition 2.6) for the harmonic functions. In the last section, we are going to show that the Weierstrass division theorem does not hold true for the rings of germs of real analytic functions at \(0\in\mathbb{R}^n\) which are definable in the ominimal structure \((\overline{\mathbb{R}}, x^{\alpha_1},\ldots,x^{\alpha_p})\) where \(\alpha_1,\ldots,\alpha_p\) are irrational real numbers.
Keywords: ominimal structures, pluriharmonic function, Weierstrass division theorem
Language: English
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For citation: Berraho, M. Pluriharmonic Definable Functions in Some oMinimal Expansions of the Real Field, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 3540.
DOI 10.46698/w980545678091g ← Contents of issue 
 

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