When the purpose is to describe the solution set of S in the algebraic closure of its coefficient field, those simpler systems are regular chains.
The Characteristic Set Method is the first factorization-free algorithm, which was proposed for decomposing an algebraic variety into equidimensional components.
Moreover, the Author, Wen-Tsun Wu, realized an implementation of this method and reported experimental data in his 1987 pioneer article titled "A zero structure theorem for polynomial equations solving".
[1] To put this work into context, let us recall what was the common idea of an algebraic set decomposition at the time this article was written.
In order to lead to a computer program, this algorithm specification should prescribe how irreducible components are represented.
In the 1980s, when the Characteristic set Method was introduced, polynomial factorization was an active research area and certain fundamental questions on this subject were solved recently[3] Nowadays, decomposing an algebraic variety into irreducible components is not essential to process most application problems, since weaker notions of decompositions, less costly to compute, are sufficient.
In the early 1990s, the notion of a regular chain, introduced independently by Michael Kalkbrener in 1991 in his PhD Thesis and, by Lu Yang and Jingzhong Zhang[4] led to important algorithmic discoveries.
In Kalkbrener's vision,[5] regular chains are used to represent the generic zeros of the irreducible components of an algebraic variety.
In the original work of Yang and Zhang, they are used to decide whether a hypersurface intersects a quasi-variety (given by a regular chain).
Regular chains have, in fact, several interesting properties and are the key notion in many algorithms for decomposing systems of algebraic or differential equations.
A bridge between these two notions, the point of view of Kalkbrener and that of Yang and Zhang, appears in Dongming Wang's paper.