On the Theories of Triangular Sets

Triangular sets appear under various names in many papers concerning systems of polynomial equations. Ritt (1932) introduced them as characteristic sets. He described also an algorithm for solving polynomial systems by computing characteristic sets of prime ideals and factorizing in field extensions. Characteristic sets of prime ideals have good properties but factorization in algebraic extensions is, most of the time, too costly. In order to avoid factorizations, Wu’s algorithm (Wu, 1986, 1987) computes characteristic sets of finite sets of polynomials which do not generate necessarily prime ideals. Wu’s algorithm may produce redundant decompositions of varieties and may not discover their possible emptiness. Several authors continued and improved Wu’s approach, mainly Chou and Gao (1990, 1991a, 1992), Gallo and Mishra (1990, 1991) and Wang (1992, 1993, 1995). Kalkbrener (1991) and Yang and Zhang (1994) introduced particular triangular sets, called regular chains. Taking advantage of the good properties of regular chains, Kalkbrener presented also an algorithm for decomposing a variety into unmixed-dimensional non-empty components described as regular chains. In addition, Lazard (1991a) introduced the notion of normalized triangular sets which are special regular chains and he provided a method for uniquely decomposing the solution of a polynomial system as regular zeros of such normalized triangular sets. This avoids the problems resulting from the non-canonicity of Wu’s and other decompositions. Following the work of Lazard (1992), an efficient alogrithm for solving zero-dimensional systems by means of normalized triangular sets is reported in Moreno Maza and Rioboo (1995). Wang (1998a) gave a generalization of the notion of a regular chain to pairs of polynomial sets (one set for equations and the other one for inequations) whose union is a triangular set, such pairs being called simple systems. He also presented an algorithm for solving polynomial systems by means of simple systems. The relationships between most of these algorithms and