Robust Algorithms For Generalized Pham Systems

Abstract

We discuss the complexity of robust symbolic algorithms solving a significant class of zero-dimensional square polynomial systems with rational coefficients over the complex numbers, called generalized Pham systems, which represent the class of zero-dimensional homogeneous complete-intersection systems with "no points at infinity". Our notion of robustness models the behavior of all known universal methods for solving (parametric) polynomial systems avoiding unnecessary branchings and allowing the solution of certain limit problems. We first show that any robust algorithm solving generalized Pham systems has complexity at least polynomial in the Bézout number of the system under consideration. Then we exhibit a robust probabilistic algorithm which solves generalized Pham systems with quadratic complexity in the Bézout number of the input system. The algorithm consists in a series of homotopies deforming the input system into a system which is "easy to solve", together with a "projection algorithm" which allows to move the solutions of the known instance to the solutions of an arbitrary instance along the parameter space.

Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with large absorption

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Efficient approximation of the solution of certain nonlinear reaction–diffusion equations with small absorption

“…We observe that the family of systems (5) has typically an exponential number O(p n ) of complex solutions ( [20]), and hence it is ill conditioned from the point of view of its solution by the so-called robust universal algorithms (cf. [21], [22], [23]). An example of such algorithms is that of general continuation methods (see, e.g., [24]).…”