Definition Let k be a field. Let R = k[[X_1, .., X_n, Y]] be the formal power series ring over k. Let f be an element of R. We say f is regular of order s with respect to Y if it satisfies the following condition. 1) f(0,..,0, Y) is not zero. 2) Consider f(0,..,0, Y) as a formal power series of one variable Y. Then s is the least integer such that Y^s has non-zero coefficient in f(0,..,0, Y).
Theorem (Weierstrass's Preparation Theorem) Let k be a field. Let R = k[[X_1, .., X_n, Y]] be the formal power series ring over k. Let f be an element of R, regular of order s with respect to Y. Then f can be uniquely expressed in the form: f = u(Y^s + h_(s-1) Y^(s-1) + ... + h_1 Y + h_0), where u is an invertible element of R, i.e. u(0, ... ,0) is not 0 and each h_i is an element of k[[X_1, .., X_n]]. Moreover, h_i(0,..,0) = 0 for all i.
