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On the Weierstrass Preparation Theorem

Volume 123 / 2019

Azimbay Sadullaev Annales Polonici Mathematici 123 (2019), 473-479 MSC: 32A60, 32U05, 32U15. DOI: 10.4064/ap181116-18-5 Published online: 10 October 2019

Abstract

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 $f ( {{z}^{0}},w )\not \equiv 0.$

Authors

  • Azimbay SadullaevNational University of Uzbekistan
    University St. 4
    100174 Tashkent, Uzbekistan
    e-mail

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