Duality and Multiplicities in Polynomial System Solving

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on different models for describing solutions, we consider suitable representations of a multiple root, or more precisely suitable descriptions of the primary component of the system at a root. We analyse the complexity of nding the representations. We also discuss the current approach to the representation of real roots. 1 Introduction When solving a 0-dimensional system of polynomial equations it is often satisfactory to know the zeroes of the system in such a way, that one can perform arithmetical operations on the coordinates of each root. One could however be interested also in the multiplicity of each root, not just in the weak \arithmetical" sense of simple, double, triple, etc. root, but in the stronger \algebraic" sense of giving a suitable description of the primary component at a root of the ideal deening the solution set of the system. The aim of this paper is to discuss a representation of the \algebraic" multiplicity and ways of computing it. There is of course a stream of research about methods for solving systems of equations (for a survey we refer to L93]) and also a "reeection" about the "meaning" of solving a system. The philosophy essentially goes back to Kro-necker: a system is solved, if each root is represented in a way which allows to perform any arithmetical operations over the arithmetical expressions of its coordinates (the operations include, in the real case, numerical interpolation). For instance, in the classical Kronecker method, concerning the univariate case, one is given a tower of algebraic eld extensions over the eld of rational numbers, each eld being a polynomial ring over the previous one modulo the