No abstract

Numerical algebraic geometry uses numerical data to describe algebraic varieties. It is based on numerical polynomial homotopy continuation, which is a technique alternative to the classical symbolic approaches of computational algebraic geometry. We present a package, whose primary purpose is to interlink the existing symbolic methods of Macaulay2 and the powerful engine of numerical approximate computations. The core procedures of the package exhibit performance competitive with the other homotopy continuation software.

Abstract. 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, writting them in a unified and accesible 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

We study methods for finding the solution set of a generic system in a family of polynomial systems with parametric coefficients. We present a framework for describing monodromy based solvers in terms of decorated graphs. Under the theoretical assumption that monodromy actions are generated uniformly, we show that the expected number of homotopy paths tracked by an algorithm following this framework is linear in the number of solutions. We demonstrate that our software implementation is competitive with the existing state-of-the-art methods implemented in other software packages.

Given an approximation to a multiple isolated solution of a polynomial system of equations, we have provided a symbolic-numeric deflation algorithm to restore the quadratic convergence of Newton's method. Using first-order derivatives of the polynomials in the system, our method creates an augmented system of equations which has the multiple isolated solution of the original system as a regular root.In this paper we consider two approaches to computing the "multiplicity structure" at a singular isolated solution. An idea coming from one of them gives rise to our new higher-order deflation method. Using higherorder partial derivatives of the original polynomials, the new algorithm reduces the multiplicity faster than our first method for systems which require several first-order deflation steps.We also present an algorithm to predict the order of the deflation.