Abstract. The constructive commutative algebra is very useful in the kinematical analysis of the mechanisms because a large class of systems can be described using polynomial equations. We show that one can analyze quite complicated systems using a sort of divide and conquer strategy to decompose the system, and hence the configuration space, into simpler parts. The key observation is that it seems that typically systems indeed have a lot of distinct components, but usually only one of them is physically relevant. Hence if one finds the equations describing the component of interest the analysis of this system can be surprisingly simple compared to the original system. In particular typically the possible singularities of the original system disappear when one restricts the attention to the relevant component. On the technical side we show that some basic constraints used to define joints in 3 dimensional mechanisms can be decomposed to simpler parts. This has significant practical consequences because using these fundamental decompositions when writing the equations for complicated mechanisms decreases dramatically the complexity of the required computations.