Counting solutions of a polynomial system locally and exactly

In this paper, we propose a symbolic-numeric algorithm to count the number of solutions of a zero-dimensional square polynomial system within a local region. We show that the algorithm succeeds under the condition that the region is sufficiently small and well-isolating for a k-fold solution z of the system. In our analysis, we derive a bound on the size of the region that guarantees success. We further argue that this size depends on local parameters such as the norm and multiplicity of z as well as the distances between z and all other solutions. Efficiency of our method stems from the fact that we reduce the problem of counting the roots of the original system to the problem of solving a truncated system of degree k. In particular, if the multiplicity k of z is small compared to the total degrees of the original polynomials, our method considerably improves upon known complete and certified methods. We see a series of applications of our approach. When combined with a numerical solver in the fashion of an a posteriori certification step, we obtain a certified and reliable method for solving polynomial systems while profiting both from the efficiency of the numerical algorithm and the reliability of the symbolic approach. An alternative application results from incorporating our algorithm as inclusion predicate into an elimination method. For the special case of bivariate systems, we experimentally show that this approach leads to a significant improvement over an existing state-of-the-art elimination method.