The Newton Procedure for several variables

Abstract

Let us consider an equation of the formwhere m > 1, x = (x 1 , .

Support of Laurent series algebraic over the field of formal power series

“…Our first main result, that will be very useful in the sequel, is a general construction of algebraically closed fields containing the field K((x)). In particular it generalizes and unifies the previous constructions given in [2,8,15,26]. This result is the following one (see Section 3 for the definition of a continuous positive order -but essentially this is a total order on R n compatible with the addition and such that the elements of R 0 n are non-negative):…”

“…Let us mention that the proof of this theorem is a direct consequence of a very nice result of Rayner [20] that has been proven twenty years before the works [2,8,15,26].…”

“…. , u n ∈ Q n are Q-linearly independent, the main result in [SV11] is a particular case of Theorem 4.5.…”

“…When n ≥ 2 there are several descriptions of algebraically closed fields containing K((x)) [Mc95,Go00,AI09,SV11]. The elements of these fields are Puiseux series whose support is included in a translated strongly convex rational cone containing R ≥0…”

Asymptotic Behavior of a Parametric Algebraic Surface

“…x y z w ∈  Furthermore, 2 t    represents the field of formal Puiseux series in the variable t 2 (see e.g., [18,19], Section 2.5 in [3,20,21], Chapter 4 (Section 2) in [4]), and let…”