Algorithms for computing triangular decomposition of polynomial systems (Q413407)

A \textit{triangular decomposition} of a system of polynomial equations is a set of simpler polynomial systems such that a solution of the initial system is a solution of one of the new systems, and vice versa. The roots of this subject trace back to the seminal works of \textit{W. T. Wu}, and specially to the paper \textit{A zero structure theorem for polynomial equations solving} in 1987 [summary in: Lect. Notes Comput. Sci. 378, 44 (1989; Zbl 1209.12012)]. In the paper under review, the authors give an overview of the key ideas in order to describe the algorithms to compute a triangular decomposition for a given system. In this direction, they propose an incremental algorithm based on the notion of \textit{regular GCD} (introduced by Moreno Maza). However, for the computation of regular GCD, they employ the specialization of property of subresultants to speed-up the computations. It should be noted that the algorithms of this paper have been implemented in the library \texttt{RegularChain} of \textsc{Maple} and it is now one of the back engines of \textsc{Maple}'s \texttt{solve} command.