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On the Weierstrass preparation theorem. (English) Zbl 1439.32019

Summary: The well-known Weierstrass Preparation Theorem states that if \(f( z,w)\) is holomorphic at a point \(( z^0,w^0)\in \mathbb{C}_z^n\times{\mathbb{C}}_w\) and \(f( z^0,w^0)=0\), but \(f (z^0,w) \not \equiv 0\), then in some neighborhood \(U=V\times W\) of this point \(f\) is represented as \[f(z,w)= [{{ (w-{{w}^0})}^m}+{{c}_{m-1}} (z){{ (w-{{w}^0})}^{m-1}}+\dots +{{c}_0}(z)]\varphi (z,w),\] where \({c}_j (z) \in \mathcal{O}(V)\), \({c}_j (z^0)=0\) for \(j=0,1,\ldots ,m-1\) and \(\varphi \in \mathcal{O}(U), \varphi (z,w)\neq 0\). In this paper, a global multidimensional (in \(w)\) analogue of this theorem is proved without the condition \(f ({{z}^0},w)\not \equiv 0\).

MSC:

32A60 Zero sets of holomorphic functions of several complex variables
32U05 Plurisubharmonic functions and generalizations
32U15 General pluripotential theory
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References:

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