 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.