On the Betti numbers of sign conditions

Let R be a real closed field and let Q and P be finite subsets of R[X 1 ,...,X k ] such that the set P has s elements, the algebraic set Z defined by Λ Q ∈ Q Q = 0 has dimension k' and the elements ofQ and P have degree at most d. For each 0 0 or P < 0 for P ∈ P, is bounded by Σ k' i=0 Σ k'-i j=0 (sj)6 j d(2d-1) k-1 . making the bound s k' O(d) k more precise.