The Abhyankar–Jung Theorem

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