Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods

In this paper, we are concerned with the following problem: given a finite family F of multivariate polynomials over a field k with ordered variables x1 < x2 < · · · < xn, we want to describe the affine variety V(F ) (i.e. the common zeroes of F over an algebraic closure of k). Such a description is usually provided by a finite family {T1, . . . , Tr} of polynomial sets with particular properties, a relation between the Ti and F , and an algorithm to compute the Ti from F . A well-developed method since Buchberger (1965) is the following: given an ordering on the monomials, choose for T1 a Gröbner basis of the ideal generated by F and compute it by the Buchberger’s algorithm. Following the work of Ritt (1932, 1966), Wu (1986) introduced another way of solving algebraic systems which is the one we are concerned with in this paper. In this case each Ti is a polynomial set such that two distinct polynomials in Ti have distinct greatest variables. Such a Ti is called a triangular set. A point ζ ∈ V(Ti) is called regular if for every p ∈ Ti the point ζ does not cancel the initial of p (that is the leading coefficient of p regarded as a univariate polynomial in its greatest variable). Then, in Wu’s method, the variety V(F ) is the union of the regular zeroes of the Ti and this decomposition can be computed by Wu’s CHRST-REM algorithm (Wu, 1987). This method has been investigated in many papers. Among them: (Chou, 1988; Chou and Gao, 1990, 1992; Gallo and Mishra, 1990; Wang, 1992, 1995). Wu’s method is efficient for geometric problems where the degenerate solutions are not interesting. For general problems it seems to be difficult to obtain an efficient implementation and this method may produce superfluous triangular sets. Wu’s algorithm, like Buchberger’s, depends on many choices; moreover, its result is not uniquely defined.