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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 $K=\mathbb{C}(x,y,z)/(f)$ 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$variables over an algebraically closed field of characteristic zero. 1 The original papers are accessible online: Jung's paper is devoted exactly to this result, whereas Abhyankar's gives a more contextualized explanation (his was an unsuccessful attempt to pass to characteristic$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.