We show that every quasi-ordinary Weierstrass polynomial P (Z) = Z + a1(X)Z d−1 + · · ·+ ad(X) ∈ K[[X]][Z], X = (X1, . . . , Xn), over an algebraically closed field of characterisic zero K, such that a1 = 0, is ν-quasi-ordinary. That means that if the discriminant ∆P ∈ K[[X]] is equal to a monomial times a unit then the ideal (a i (X))i=2,...,d is monomial and generated by one of a i (X). We us...