Question

I can find many modification of the Jung-Abhyankar 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?

Answer 1

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, 289-314 (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 Abhyankar-Jung 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.

Answer 2

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