Given a system of polynomial equations in several complex variables, is there a way to determine if the solution space has disconnected components using some computational algebra system like Mathematica, Magma, Macauly2, Singular, etc?
In case it is helpful: The system of equations I am interested in is small enough that I can calculate the Groebner basis over the rationals, the solution space is not zero-dimensional, and (after a long compute) Singular was able to give the prime decomposition of the ideal generated by the system of polynomial equations.
My current plan is to now check if the parts of the decomposition share share solutions by checking if the basis for the union of their generators does not collapse to 1. But I don't know how to check if an individual part has disconnected components or not. Maybe I am over complicating this, and there is an easier method to check connectedness? Is it sufficient to check if each polynomial in the basis over the rationals is irreducible over the complex numbers as well?