Characteristic polyhedra of singularities

Abstract

T h e very basic idea o f this work is that of Newton polygon. Just to illuminate this view-point, let us take an irreducible plane curve X through the origin defined by an equation f(x , y )= 0 , say over an algebraically closed field k. Let y b e the multiplicity o f X a t the origin 0, j. e., the order o f f a t 0. Then we choose the parameters in such a way that the coefficient of y ' is not zero in f(x , y ) . Let f =S,, j c x'yfi be the power series expansion of f. We map each term cd x y ' to the point (i/v, j/v) of the Cartesian plane R 2 . Let E be the set of those points corresponding to nonzero terms of f , and F the convex closure of the union U , E E v+ R ( The part of the boundary o f F not contained in the tw o axes is called the N ew ton poly gon of f with respect to the parameters x and y- A s we have c0 "# 0 and c 15 = 0 f o r all i + j < v , the first segment of the Newton polygon has (0 ,1 ) as its left end point.