I have a large system of polynomial equations and am trying to construct an irreducible decomposition of the corresponding variety. It's large enough that feeding it to the standard radical/primary decomposition algorithm in Singular is computationally prohibitive.
However, many of the equations in the system factor. If I introduce an additional assumption that there are no zero divisors, then a polynomial that factors (say $ab$) allows me to split the system into two systems, one containing $a=0$ and one containing $b=0$. Knowing that $a=0$ in the first of these systems might produce additional factorization, and additional splitting. This kind of pre-processing might produce a set of simpler systems that could then be fed to Singular, or a similar program, to complete the final decomposition.
Is anybody aware of such an algorithm? Has it been published anywhere?