On Multiplicities in Polynomial System

This paper deals with the description of the solutions of zero dimensional systems of polynomial equations. Based on diierent 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 and of algorithms which perform transformations between the diierent representations. Introduction When solving a system of polynomial equations (which in this paper will always have a 0-dimensional set of zeroes and will be called " 0-dimensional system"), it is often satisfactory to know the zeroes of the system, in such a way to be able to perform arithmetical operations on the coordinates of each root. There is of course a stream of research about methods for solving systems of equations (for a survey we refer to L93]). There is also a "reeection" about the "meaning" of solving a system, taking place within some research groups more interested in eeective methods for algebraic geometry. The philosophy essentially goes back to Kronecker: a system is solved if each root is represented in such a way to allow to perform any arithmetical operations over the arithmetical expressions of its coordinates (the operations including, 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 of the eld of rational numbers, each eld being a polynomial ring over the previous one modulo the ideal generated by a single polynomial and each root is represented by an element in such elds. The main eeort of the actual research is devoted to eeective techniques for representing roots of a systems and allowing eecient arithmetical operations over their expressions. In this context 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 suitable approaches to this question based on diierent models for computing solutions (i.e. without multiplicity). The \arithmetical" multiplicity of a primary at the origin can be easily computed since it is read from the leading term of the Hilbert polynomial but this doesn't give a …