The Abhyankar–Jung theorem for excellent henselian subrings of formal power series

Given an algebraically closed field K of characteristic zero, we present the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring K[[z]]. In particular, examples include the local rings which form a Weierstrass system over the field K. The Abhyankar–Jung theorem may be regarded as a higher dimensional counterpart of the Newton–Puiseux theorem. It asserts that the roots of a Weierstrass (formal or convergent) polynomial over an algebraically closed field of characteristic zero with discriminant being a normal crossing are fractional (formal or convergent) series. The first proof for the case of two complex variables was due to H.W. Jung [6]. The general result for the algebroid case (for several variables and an arbitrary ground field) was established by S.S. Abhyankar [1] by means of purely algebraic methods, namely, some properties of the Galois group of the polynomial under study. The methods of Jung and Abhyankar are described e.g. in [8]. The classical proofs of the Newton–Puiseux theorem applied Newton’s algorithm to compute, term by term, the fractional series (called Puiseux series) arising as t-roots of an algebraic equation f(x, t) = 0. This algorithm had been invented in [11], ”Methodus fluxionum et serierum infinitorum” (see also [19]). It consists in computation using the so-called Newton polygon, determined by the exponents of a given polynomial. 2010 MSC: 13F25, 13F40.