Generic Regular Decompositions for Parametric Polynomial Systems

TL;DR

This paper extends the concepts of generic regular decomposition and RDU varieties to parametric polynomial systems beyond zero-dimensional cases, providing an algorithm for stable solutions representation.

Contribution

It introduces a new algorithm for computing GRDs and RDU varieties for general parametric systems, broadening the applicability of previous zero-dimensional methods.

Findings

Algorithm successfully computes GRDs and RDU varieties.

Decomposition remains stable outside the RDU variety.

Implemented and tested on benchmark problems.

Abstract

This paper presents a generalization of our earlier work in [19]. In this paper, the two concepts, generic regular decomposition (GRD) and regular-decomposition-unstable (RDU) variety introduced in [19] for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in [19]. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in [19] are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.