On the Chordality of Polynomial Sets in Triangular Decomposition in Top-Down Style

Abstract

In this paper the chordal graph structures of polynomial sets appearing in triangular decomposition in top-down style are studied when the input polynomial set to decompose has a chordal associated graph. In particular, we prove that the associated graph of one specific triangular set computed in any algorithm for triangular decomposition in top-down style is a subgraph of the chordal graph of the input polynomial set and that all the polynomial sets including all the computed triangular sets appearing in one specific simply-structured algorithm for triangular decomposition in top-down style (Wang's method) have associated graphs which are subgraphs of the the chordal graph of the input polynomial set. These subgraph structures in triangular decomposition in top-down style are multivariate generalization of existing results for Gaussian elimination and may lead to specialized efficient algorithms and refined complexity analyses for triangular decomposition of chordal polynomial sets.