Generic Regular Decompositions for Generic Zero-Dimensional Systems

TL;DR

This paper introduces generic regular decompositions and RDU varieties for zero-dimensional systems, providing an algorithm that computes these decompositions efficiently and stably for generic parameter values.

Contribution

It proposes a novel algorithm for computing generic regular decompositions and RDU varieties simultaneously, based on subresultant chains and weakly relatively simplicial decomposition.

Findings

Algorithm computes decompositions efficiently on benchmarks.

Decomposition is stable outside the RDU variety.

Empirical results demonstrate good performance.

Abstract

Two new concepts, generic regular decomposition and regular-decomposition-unstable (RDU) variety for generic zero-dimensional systems, are introduced in this paper and an algorithm is proposed for computing a generic regular decomposition and the associated RDU variety of a given generic zero-dimensional system simultaneously. The solutions of the given system can be expressed by finitely many zero-dimensional regular chains if the parameter value is not on the RDU variety. The so called weakly relatively simplicial decomposition plays a crucial role in the algorithm, which is based on the theories of subresultant chains. Furthermore, the algorithm can be naturally adopted to compute a non-redundant Wu's decomposition and the decomposition is stable at any parameter value that is not on the RDU variety. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Empirical results are also presented to show the good performance of the algorithm.