On the number of cells defined by a family of polynomials on a variety

Let R be a real closed field and V a variety of real dimension k' which is the zero set of a polynomial Q is an element of R[X(1),...,X(k)] of degree at most d. Given a family of s polynomials P = {P-1,..., P-s} subset of R[X(1),...,X(k)] where each polynomial in P has degree at most d, we prove that the number of cells defined by P over V is ((K)'(s))(O(d))(k). Note that the combinatorial part of the bound depends on the dimension of the variety rather than on the dimension of the ambient space.