A Polyhedral Homotopy Algorithm For Real Zeros

We design a homotopy continuation algorithm, that is based on numerically tracking Viro's patchworking method, for finding real zeros of sparse polynomial systems. The algorithm is targeted for polynomial systems with coefficients satisfying certain concavity conditions. It operates entirely over the real numbers and tracks the optimal number of solution paths. In more technical terms; we design an algorithm that correctly counts and finds the real zeros of polynomial systems that are located in the unbounded components of the complement of the underlying A-discriminant amoeba.Comment: some cosmetic changes are done and a couple of typos are fixed to improve readability, mathematical contents remain unchange

Zeros of Lacunary T ype of Polynomials

In this paper we use matrix methods and Gereshgorian disk Theorem to present some interesting generalizations of some well-known results concerning the distribution of the zeros of polynomial. Our results include as a special case some results due to A .Aziz and a result of Simon Reich-Lossa

Globally Convergent Parallel Algorithm for Zeros of Polynomial Systems

Certain classes of nonlinear systems of equations, such as polynomial systems, have properties that make them particularly amenable to solution on distributed computing systems. Some algorithms, considered unfavorably on a single processor serial computer, may be excellent on a distributed system. This paper considers the solution of polynomial systems of equations via a globally convergent homotopy algorithm on a hypercube. Some computational results are reported

Singular Zeros of Polynomial Systems

International audienceSingular zeros of systems of polynomial equations constitute a bottleneck when it comes to computing, since several methods relying on the regularity of the Jacobian matrix of the system do not apply when the latter has a non-trivial kernel. Therefore they require special treatment. The algebraic information regarding an isolated singularity can be captured by a finite, local basis of differentials expressing the multiplicity structure of the point. In the present article, we review some available algebraic techniques for extracting this information from a polynomial ideal. The algorithms for extracting the, so called, dual basis of the singularity are based on matrix-kernel computations, which can be carried out numerically, starting from an approximation of the zero in question. The next step after obtaining the multiplicity structure is to deflate the root, that is, construct a new system in which the singularity is eliminated. Having a deflated system allows to refine the solution fast and to high accuracy, since the Jacobian matrix is regular and all the usual machinery, e.g. Newton's method or existence and unicity criteria may be applied. Standard verification methods, based e.g. on interval arithmetic and a fixed point theorem, can then be employed to certify that there exists a unique perturbed system with a singular root in the domain