On the Weierstrass Preparation Theorem

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

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