The Abhyankar-Jung Theorem

We show that every quasi-ordinary Weierstrass polynomial P(Z) = z(d) + a(1) (X)z(d-1) + ... + a(d)(X) is an element of K[[X]][Z], X = (X-1, ... , X-n), over an algebraically closed field of characteristic zero such that a(1) = 0, is v-quasi-ordinary. That means that if the discriminant Delta(p) is an element of K[[X]] is equal to a monomial times a unit then the ideal (a(i)(d!/i) (X))(i=2 , ... , d) is monomial and generated by one of a(i)(d!/i) (X). We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of K[[X]] and the function germs of quasi-analytic families. (C) 2012 Elsevier Inc. All rights reserved.