Weierstrass polynomial

Summary

A Weierstrass polynomial is a monic polynomial in one variable $z_n$ whose other coefficients are holomorphic germs in $z_1, \dots, z_{n-1}$ vanishing at the origin. The Weierstrass preparation theorem says that every holomorphic germ that does not vanish identically on the $z_n$-axis is a unit times such a polynomial. This reduces many local questions about holomorphic functions of $n$ variables to questions about polynomials over the ring of germs in $n - 1$ variables.