Stability of Triangular Decomposition and Comprehensive Triangular Decomposition

TL;DR

This paper introduces the concept of decomposition-unstable varieties in parametric polynomial systems, analyzes the stability of various triangular decomposition methods, and proposes a new algorithm for comprehensive triangular decomposition that is efficient and implemented in Maple.

Contribution

It defines weakly comprehensive triangular decomposition and develops a hierarchical, self-adaptive algorithm for computing comprehensive triangular decompositions, improving efficiency over existing methods.

Findings

The new algorithm is faster or comparable in speed to existing Maple packages.

The concept of decomposition-unstable varieties helps analyze the stability of decomposition methods.

Experimental results demonstrate the effectiveness of the proposed algorithm.

Abstract

A new concept, decomposition-unstable (DU) variety of a parametric polynomial system, is introduced in this paper and the stabilities of several triangular decomposition methods, such as characteristic set decomposition, relatively simplicial decomposition and regular chain decomposition, for parametric polynomial systems are discussed in detail. The concept leads to a definition of weakly comprehensive triangular decomposition (WCTD) and a new algorithm for computing comprehensive triangular decomposition (CTD) which was first introduced in [4] for computing an analogue of comprehensive Groebner systems for parametric polynomial systems. Our algorithm takes advantage of a hierarchical solving strategy and a self-adaptive order of parameters. The algorithm has been implemented with Maple 15 and experimented with a number of benchmarks from the literature. Comparison with the Maple package RegularChains, which contains an implementation of the algorithm in [4], is provided and the results illustrate that the time costs by our program for computing CTDs of most examples are no more than those by RegularChains.