Over a complete local ring, a power series whose first unit coefficient has index is a unit times a unique monic distinguished polynomial of degree . Consequently its sufficiently small zeros, counted with multiplicity, are the zeros of that polynomial.
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The Weierstrass Preparation Theorem is a fundamental result in complex analysis and algebraic geometry concerning the behavior of holomorphic functions near a point where they have a zero. It is particularly important in the study of local properties of holomorphic functions and their singularities.