Variations on inversion theorems for Newton-Puiseux series

TL;DR

This paper revisits classical inversion theorems for Newton-Puiseux series, providing new proofs and generalizations to multivariable cases, enhancing understanding of the relationship between series expansions when solving algebraic equations.

Contribution

The authors present two new proofs of a forgotten inversion theorem and extend it to equations with multiple variables, broadening its applicability.

Findings

New proofs of the classical inversion theorem

Generalization to multivariable equations

Enhanced understanding of characteristic exponents and coefficients

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.