Weierstrass polynomial

Definition.

A function $W:{ℂ}^n\to ℂ$ of the form

$$W(z_1,\mathrm{\dots },z_n)=z_n^m+\sum _{j=1}^{m-1}{a}_j(z_1,\mathrm{\dots },z_{n-1})z_n^j,$$

where the $a_j$ are holomorphic functions in a neighbourhood of the origin, which vanish at the origin,is called a Weierstrass polynomial.

Any codimension 1 complex analytic subvariety of ${ℂ}^n$ can be written as the zero setof a Weierstrass polynomial using the Weierstrass preparation theorem. This in general cannot be done for higher codimension.