On a real analogue of Bezout inequality and the number of connected components of sign conditions

Let R be a real closed field and Q1, , Ql∈R[X1, ,Xk] such that for each i,1≤ i≤ l, deg (Qi)≤di. For 1≤ i≤ l, denote by Qi={Q1, , Qi}, Vi the real variety defined by Qi, and ki an upper bound on the real dimension of Vi (by convention V0=R k and k0= k). Suppose also that 2≤ d1≤ d2≤ 1 k+1 d3≤ 1 (k+1)2 d4≤ ≤ 1 (k+1)l−3 dl−1≤ 1 (k+1)l−2 dl, and that l≤k. We prove that the number of semi-algebraically connected components of Vl is bounded by O(k) (