 Weierstrass preparation theorem

Over a complete local ring, a power series whose first unit coefficient has index $d$ is a unit times a unique monic distinguished polynomial of degree $d$. Consequently its sufficiently small zeros, counted with multiplicity, are the zeros of that polynomial.

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.