The Newton Procedure for several variables

Abstract

Let us consider an equation of the form P(x, z) = zm + w1(x)zm−1 + · · · + wm−1(x)z + wm(x) = 0, where m>1, n>1, x=(x1⋯xn) is a vector of variables, k is an algebraically closed field of characteristic zero, View the MathML source and wm(x)≠0. We consider representations of its roots as generalized Puiseux power series, obtained by iterating the classical Newton procedure for one variable. The key result of this paper is the following: Theorem 1.The iteration of the classical Newton procedure for one variable gives rise to representations of all the roots of the equation above by generalized Puiseux power series in x1/d, d∈Z>0, whose supports are contained in an n-dimensional, lex-positive strictly convex polyhedral cone (see Section 5). We must point out that the crucial result is not the existence of these representations, which is a well-known fact; but the fact that their supports are contained in such a special cone. We achieve the proof of this theorem by taking a suitable affine chart of a toric modification of the affine space.