The Abhyankar-Jung Theorem

TL;DR

This paper provides a constructive proof of the Abhyankar-Jung Theorem for quasi-ordinary Weierstrass polynomials over algebraically closed fields of characteristic zero, extending to Henselian local rings and quasi-analytic families.

Contribution

It proves that quasi-ordinary Weierstrass polynomials are $ u$-quasi-ordinary and offers a constructive proof of the Abhyankar-Jung Theorem applicable in broader contexts.

Findings

Quasi-ordinary Weierstrass polynomials are $ u$-quasi-ordinary.

Constructive proof of Abhyankar-Jung Theorem for Henselian local rings.

Applicable to function germs of quasi-analytic families.

Abstract

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 $\K$, and satisfying $a_1=0$, is $\nu$-quasi-ordinary. That means that if the discriminant $\Delta_P \in \K[[X]]$ is equal to a monomial times a unit then the ideal $(a_i^{d!/i}(X))_{i=2,...,d}$ 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 $\K[[X]]$ and the function germs of quasi-analytic families.