# Reformulation of a semialgebraic set, without quantifiers

According to wikipedia:

The Tarski–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 Tarski–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.

For the record: I couldn't get qepcad to make a projection of codimension 2 using even 4 points, but I did get some significant results by restricting to a three dimensional space, so that qepcad only projects along lines.