Certified Numerical Homotopy Tracking

Given a homotopy connecting two polynomial systems we provide a rigorous algorithm for tracking a regular homotopy path connecting an approximate zero of the start system to an approximate zero of the target system. Our method uses recent results on the complexity of homotopy continuation rooted in the alpha theory of Smale. Experimental results obtained with the implementation in the numerical algebraic geometry package of Macaulay2 demonstrate the practicality of the algorithm. In particular, we confirm the theoretical results for random linear homotopies and illustrate the plausibility of a conjecture by Shub and Smale on a good initial pair. The numerical homotopy continuation methods are the backbone of the area of numerical algebraic geometry; while this area has a rigorous theoretic base, its existing software relies on heuristics to perform homotopy tracking. This paper has two main goals: • On one hand, we intend to overview some recent developments in the analysis of complexity of polynomial homotopy continuation methods with the view towards a practical implementation. In the last years, there has been much progress in the understanding of this problem. We hereby summarize the main results obtained, writing them in a unified and accessible way. • On the other hand, we present for the first time an implementation of a certified homotopy method which does not rely on heuristic considerations. Experiments with this algorithm are also presented, providing for the first time a tool to study deep conjectures on the complexity of homotopy methods (as Shub & Smale’s conjecture discussed below) and illustrating known – yet somehow surprising – features about these methods, as Date: June 17, 2011. C. Beltrán. Departamento de Matemáticas, Estad́ıstica y Computación, Universidad de Cantabria, Spain ([email protected]). Partially supported by MTM2007-62799 and MTM2010-16051, Spanish Ministry of Science (MICINN).. Anton Leykin. School of Mathematics, Georgia Tech, Atlanta GA, USA ([email protected]). Partially supported by NSF grant DMS-0914802.