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) \in \K[[X]][Z] $,$X=(X_1,…, X_n)$, over an algebraically closed field of characterisic zero $\text{K}$, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant $\Delta_P \in \K[[X]]$isequaltoamonomialtimesaunitthentheideal$(a_i^{d!/i}(X))_{i=2,…,d}$ is principal and generated by a monomial. We use this result to give a constructive proof of the Abhyankar-Jung Theorem that works for any Henselian local subring of $\text{K}[[X]]$ and the function germs of quasi-analytic families.