Hierarchical Comprehensive Triangular Decomposition

TL;DR

This paper introduces a hierarchical strategy for computing comprehensive triangular decompositions (CTD) of parametric polynomial systems, dividing the parametric space into sub-spaces to improve computational efficiency and obtain partial solutions.

Contribution

The paper proposes a novel hierarchical approach for CTD computation that enhances efficiency and provides partial solutions by dividing the parametric space into sub-spaces.

Findings

The hierarchical strategy can compute CTDs more efficiently on benchmarks.

Partial solutions are obtainable over some parametric sub-spaces.

Experimental results show competitive performance compared to existing methods.

Abstract

The concept of comprehensive triangular decomposition (CTD) was first introduced by Chen et al. in their CASC'2007 paper and could be viewed as an analogue of comprehensive Grobner systems for parametric polynomial systems. The first complete algorithm for computing CTD was also proposed in that paper and implemented in the RegularChains library in Maple. Following our previous work on generic regular decomposition for parametric polynomial systems, we introduce in this paper a so-called hierarchical strategy for computing CTDs. Roughly speaking, for a given parametric system, the parametric space is divided into several sub-spaces of different dimensions and we compute CTDs over those sub-spaces one by one. So, it is possible that, for some benchmarks, it is difficult to compute CTDs in reasonable time while this strategy can obtain some "partial" solutions over some parametric sub-spaces. The program based on this strategy has been tested on a number of benchmarks from the literature. Experimental results on these benchmarks with comparison to RegularChains are reported and may be valuable for developing more efficient triangularization tools.