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

Let R be a real closed field and a variety of real dimension k ′ which is the zero set of a polynomial Q ∈ R [ X 1 ,…, X k ] of degree at most d . Given a family of s polynomials = { P 1 ,…, P s }⊂ R [ X 1 ,…, X k ] where each polynomial in has degree at most d , we prove that the number of cells defined by over is ( 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.