On approximate triangular decompositions in dimension zero

Triangular decompositions for systems of polynomial equations with n variables, with exact coefficients are well-developed theoretically and in terms of implemented algorithms in computer algebra systems. However there is much less research about triangular decompositions for systems with approximate coefficients. In this paper we discuss the zero-dimensional case, of systems having finitely many roots. Our methods depend on having approximations for all the roots, and these are provided by the homotopy continuation methods of Sommese, Verschelde and Wampler. We introduce approximate equiprojectable decompositions for such systems, which represent a generalization of the recently developed analogous concept for exact systems. We demonstrate experimentally the favourable computational features of this new approach, and give a statistical analysis of its error.