Separating families for semi-algebraic sets

A new invariantp(V) is defined for real algebraic varietiesV which measures the complexity of semi-algebraic sets inV.p(V) is the least integer such that every semi-algebraic setS ?-V can be separated from its compliment byp(V) polynomials. This is a very natural invariant to consider. Using results of Bröcker 4–8] and generalizations of Bröcker’s results found in 16,17], upper bounds forp(V) are computed. The proof is simpler than the proof of similar results in 5–9],15–18] since the complicated local-global formula for the stability index and the various pasting techniques are not needed. Lower bounds forp(V) are also computed in some special cases, the technique here being to first study the corresponding invariantp(X, G) for a finite space of orderings (X, G) 13,14].