I can find many modification of the JungAbhyankar theorem. I can even find a new proof of the theorem (by K. Kiyek and J. L. Vicente). But I cannot find the original statement. Does any one know which paper/book is it in? Where can I access it?
2 Answers
The original papers are accessible online:
 H. W. E. Jung, Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umgebung einer Stelle x = a, y = b. Journal für die reine und angewandte Mathematik 133, 289314 (1908)
 S. S. Abhyankar, On the ramification of algebraic functions. Amer. J. Math. 77 (1955), 575–592.
Jung's paper is devoted exactly to this result, whereas Abhyankar's gives a more contextualized explanation (his was an unsuccessful attempt to pass to characteristic $p$); I think the version of AbhyankarJung in Abhyankar is Theorem 3 (but it might be worth studying the paper carefully).
In Jung's statement, which I reproduce here, the field $K$ is defined as $K=\mathbb{C}(x,y)[z]/(f)$ for some irreducible polynomial $f$ (which is implicitly assumed to involve all three variables).
Man kann Funktionenpaare $u, v$ des Körpers $K$ bestimmen derart, daß $x$ und $y$ gewönliche Potenzreihen von $u$, $v$ werden, die für $u=v=0$ verschwinden, während alle anderen Funktionen von $K$ entweder gewöhnliche Potenzreihen von $u, v$ werden, oder Quotienten solcher. Eine endliche Anzahl solcher Funtionenpaare und Entwicklungen genügt, die Funktionen von $K$ für die ganze Umgebung von $x=0, y=0$ darzustellen.
My translation:
It is possible to determine pairs of functions $u, v \in K,$ such that $x$ and $y$ become usual power series in $u, v$, vanishing for $u=v=0$, while every other function in $K$ is either a usual power series in $u,v$ or a quotient of such. A finite number of such pairs and series is enough to represent all functions of $K$ in a neighborhood of $x=0, y=0$.
This seems to be equivalent, in the formulation usual in more recent papers, to the following (I use $\mathbb{C}\{x\}$ to denote convergent power series):
Let $f\in\mathbb{C}\{x,y\}[z]$ be a monic irreducible Weierstrass polynomial having a discriminant of the form $x^\alpha y^\beta u$, with $\alpha, \beta$ nonnegative integers, and $u\in \mathbb{C}\{x,y\}$ a unit. Then there exist positive integers $n, m$ such that $f$ has all its roots in $\mathbb{C}\{x^{\frac{1}{n}},y^{\frac{1}{m}}\}$.
Abhyankar considers the case of $n$ variables over an algebraically closed field of characteristic zero.

$\begingroup$ K may be not a field.I think that you mean qutient field of K $\endgroup$– gaussMar 30, 2012 at 13:54

$\begingroup$ Thanks! The notation did not make much sense, really. $\endgroup$– quimMar 30, 2012 at 16:56

$\begingroup$ Abhyankar's approach seems purely formal. Does the result tell us anything about the convergence of the fractional power series? $\endgroup$– ssquiddApr 1, 2012 at 15:09

$\begingroup$ Abhyankar's motivation (Zariski's!) was trying to generalize to positive characteristic, so he probably didn't care much for convergence. Jung does care about convergence, and I suppose the series are convergent for more variables too, one should look at modern proofs I guess. $\endgroup$– quimApr 10, 2012 at 9:21
The fractional power series solutions remain convergent in several variables. In fact, if you replace the ring of convergent power series by any Henselian ring $H$ of power series series satisfying some stability properties, then the fractional power series solution remain in the ring. See for instance http://arxiv.org/abs/1103.2559