Singularities of the Newton Mapping and The Van Der Monde Determinant

Abstract

The fact that complex polynomials can be written in two forms , k ITC(+ Zi) = e + a1(k-l + ... + ak , i=l can be used to define a mapping N(z) = a from C k to C k , known as the Newton mapping. It is clearly surj ective and turns out to have particularly useful properties regarding its symmetry, for example the analytic theorem of Newton- Let f (z1 , ... , zk) be an analytic function which is symmetric in z1 , . . . , zk, then there exists a unique analytic function g(a1, ... , ak ) such t hat f = goN.