In mathematics, and more specifically in computer algebra and elimination theory, a regular chain is a particular kind of triangular set of multivariate polynomials over a field, where a triangular set is a finite sequence of polynomials such that each one contains at least one more indeterminate than the preceding one.
Regular chains enhance the notion of Wu's characteristic sets in the sense that they provide a better result with a similar method of computation.
To fix this degenerated case, the notion of regular chain was introduced, independently by Kalkbrener (1993), Yang and Zhang (1994).
Regular chains are special triangular sets which are used in different algorithms for computing unmixed-dimensional decompositions of algebraic varieties.
Kalkbrener's original definition was based on the following observation: every irreducible variety is uniquely determined by one of its generic points and varieties can be represented by describing the generic points of their irreducible components.
Note that: (1) the content of the second polynomial is x2, which does not contribute to the generic points represented and thus can be removed; (2) the dimension of each component is 1, the number of free variables in the regular chain.
is a triangular set, if the polynomials in T are non-constant and have distinct main variables.
The quasi-component W(T) described by the regular chain T is the set difference of the varieties V(T) and V(h).
The attached algebraic object of a regular chain is its saturated ideal A classic result is that the Zariski closure of W(T) equals the variety defined by sat(T), that is, and its dimension is n − |T|, the difference of the number of variables and the number of polynomials in T. In general, there are two ways to decompose a polynomial system F. The first one is to decompose lazily, that is, only to represent its generic points in the (Kalkbrener) sense, The second is to describe all zeroes in the Lazard sense, There are various algorithms available for triangular decompositions in either sense.