The Newton Procedure for several variables

Let us consider an equation of the formP(x,z)=zm+w1(x)zm-1+⋯+wm-1(x)z+wm(x)=0,where m>1m>1, n>1n>1, x=(x1⋯xn)x=(x1⋯xn) is a vector of variables, k   is an algebraically closed field of characteristic zero, wi(x)∈k〚x〛 and wm(x)≠0wm(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/dx1/d,  d∈Z>0d∈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.