Properties of regular systems and algorithmic improvements for regular decomposition

In this paper, we study the properties of regular systems and improve the efficiency of the regular decomposition method RegSer implemented in Epsilon. We define a weaker concept which retains most properties of regular system. It can be shown that from a weak regular system one can also define a regular set and vice versa. We present an algorithm RecurWeakRegSer to decompose a given polynomial system [P,Q][P,Q] into weak regular systems. When Q≠0̸Q≠0̸, the output of RecurWeakRegSer([P,Q][P,Q]) often contains fewer components than that of RegSer([P,Q][P,Q]). This is one advantage of RecurWeakRegSer. Another one is that RecurWeakRegSer is more efficient than RegSer. This was shown by experiments that we carried out. Since it is an essential step in RegSer to compute subresultant polynomial remainder sequences (PRS), and there is some weakness in the implementation, we implement a new version of subresultant algorithm using the optimization strategy of Ducos so that the efficiency of RegSer can be improved.