### Reformulation of a semialgebraic set, without quantifiers

According to wikipedia:

The Tarskiâ€“Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities.

That's great, and my question is if it can be done in practice, in a particular case. I have read that the original Tarskiâ€“Seidenberg algorithm is of little practical use, but maybe my problem is tractable with another algorithm:

We have a finite set of points in C^2 (z_i,w_i), and I want to project onto the first C coordinate the intersection of the cones {(z,w): |z-z_i|>|w-w_i|}. It would be awesome, for example for plotting the set, if I could actually get the polynomial inequalities that define the projection. Getting only the higher dimensional strata is fine (an open set in C).

Any idea? Thanks in advance.