MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
up vote 3 down vote accepted

The original papers are accessible online:

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.

share|cite|improve this answer
K may be not a field.I think that you mean qutient field of K – gauss Mar 30 '12 at 13:54
Thanks! The notation did not make much sense, really. – quim Mar 30 '12 at 16:56
Abhyankar's approach seems purely formal. Does the result tell us anything about the convergence of the fractional power series? – ssquidd Apr 1 '12 at 15:09
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. – quim Apr 10 '12 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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.