Kronecker?s and Newton?s approaches to solving : A first comparison

Abstract

In these pages we make a first attempt to compute efficiency of symbolic and numerical analysis procedures that solve systems of multivariate polynomial equations. In particular, we compare Kronecker?s solution (from the symbolic approach) with approximate zero theory (introduced by M. Shub & S. Smale as a foundation of numerical analysis). To this purpose we show upper and lower bounds of the bit length of approximate zeros. We also introduce efficient procedures that transform local Kronecker?s solution into approximate zeros and conversely. As an application of our study we exhibit an efficient procedure to compute splitting fields and Lagrange resolvent of univariate polynomial equations. We remark that this procedure is obtained by a convenient combination of both approaches (numeric and symbolic) to multivariate polynomial solving.