## Introduction to Abhyankar-Jung Theorem

Adam Parusinski

Nice University

Abstract

Abhyankar-Jung Theorem is a multivariable generalization of Newton-Puiseux Theorem. It says that the roots of a polynomial P(Z) = Z^d+a_1 (X) Z^{d-1}+ . . . +a_d(X), where a_i (X) are complex analytic function germs of many complex variables X=(X_1,..., X_n), are convergent fractional (i.e. with positive rational exponents) power series, provided the discriminant of P is a monomial in X times an analytic unit. A similar statement holds for formal power series over an algebraically closed field K of characteristic zero. In this talk we give also a constructive proof of the latter statement by completing and old proof of Luengo. Our method can be applied to to any Henselian local subring of K[[X]] in particular to the quasi-analytic functions. (this is joint work with Guillaume Rond from Marseille)