Variations on inversion theorems for Newton-Puiseux series

Abstract

Let $f(x,y)$ be a complex irreducible formal power series without constant term. One may solve the equation $f(x,y)=0$ by choosing either $x$ or $y$ as independent variable, getting two finite sets of Newton-Puiseux series. In 1967 and 1968, Abhyankar and Zariski published proofs of an \emph{inversion theorem}, expressing the \emph{characteristic exponents} of one set of series in terms of those of the other ones. In fact, a more general theorem, stated by Halphen in 1876 and proved by Stolz in 1879, relates also the \emph{coefficients} of the characteristic terms of both sets of series. This theorem seems to have been completely forgotten. We give two new proofs of it and we generalize it to a theorem concerning equations with an arbitrary number of variables.

arXiv comment27 pages. This is the final published version. The introduction and several proofs were modified according to the recommendations of the referee. The bibliography was augmented, Mathematische Annalen. Online first on 03.12.2016

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