Multiple roots of polynomial systems

We consider a general polynomial with variable coefficients. The main result involves rational expressions in coefficients for multiple roots of the polynomial which are given in terms of resultants of the polynomial and its derivatives. We also prove similar results for a system of $n$ polynomial equations in $n$ unknowns. Basic tools we use in the proof are properties of the logarithmic Gauss mapping for discriminant set of the system and the linearization procedure. Formulas we present can be applied both in the algebra of polynomials and in problems of applied mathematics concerned with analysis of critical points of the polynomial mappings.++