Clustering Complex Zeros of Triangular Systems of Polynomials

This paper gives the first algorithm for finding a set of natural ϵ-clusters of complex zeros of a regular triangular system of polynomials within a given polybox in Cⁿ, for any given ϵ> 0. Our algorithm is based on a recent near-optimal algorithm of Becker et al. (Proceedings of the ACM on international symposium on symbolic and algebraic computation, 2016) for clustering the complex roots of a univariate polynomial where the coefficients are represented by number oracles. Our algorithm is based on recursive subdivision. It is local, numeric, certified and handles solutions with multiplicity. Our implementation is compared to with well-known homotopy solvers on various triangular systems. Our solver always gives correct answers, is often faster than the homotopy solvers that often give correct answers, and sometimes faster than the ones that give sometimes correct results.