Clustering Complex Zeros of Triangular System of Polynomials

Rémi Imbach (1) , Marc Pouget (2) , Chee Yap (1)

1 CIMS - Courant Institute of Mathematical Sciences [New York]
2 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm

This report is about finding clusters of complex solutions of triangular systems of polynomial equations. We introduce the local solution clustering problem for a system of polynomial equations, that is grouping all its complex solutions lying in an initial complex domain in clusters smaller than a given real number $\epsilon>0$, and counting the sum of multiplicities of the solutions in each clusters. For triangular systems, we propose a criterion based on the Pellet theorem to count the sum of the multiplicities of the solutions in a cluster. We also propose an algorithm for solving the local solution clustering problem for triangular systems, based on a recent near-optimal algorithm for clustering the complex roots of univariate polynomials. Our algorithm is numeric and certified. We implemented it and compared it with two homotopy solvers for randomly generated triangular systems. Our solver always give correct answers, is often faster than the homotopy solver that gives often correct answers, and sometimes faster than the one that gives sometimes correct results.

Domaines: Géométrie algorithmique [cs.CG]; Logiciel mathématique [cs.MS]

Rémi Imbach, Marc Pouget, Chee Yap. Clustering Complex Zeros of Triangular System of Polynomials. CASC 2019, 2019, Moscow, Russia. ⟨hal-01825708v2⟩