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

Let $\mathrm{R}$ be a real closed field and $Q_1, \ldots, Q_{\ell} \in \mathrm{R}[X_1, \ldots,X_k]$ such that for each $i, 1 \leq i \leq \ell$, $\deg (Q_i) \leq d_i$. For $1 \leq i \leq \ell$, denote by $\mathcal{Q}_i = \{Q_1, \ldots, Q_i \}$, $V_i$ the real bvariety defined by $\mathcal{Q}_i$, and $k_i$ an upper bound on the real dimension of $V_i$ (by convention $V_0 = ^k$ and $k_0 = k$). Suppose also that \[ 2 \leq d_1 \leq d_2 \leq \frac{1}{k + 1} d_3 \leq \frac{1}{(k + 1)^2 d_4 \leq \cdots \leq \frac{1}{(k + 1)^{\ell - 3}} d_{\ell - 1} \leq \frac{1}{(k + 1)^{\ell - 2}} d_{\ell}, \] and that $\ell \leq k$. We prove that the number of semi-algebraically connected components of $V_{\ell}$ is bounded by \[ O (k)^{2 k} \left(\prod_{1 \leq j < \ell} d_j^{k_{j - 1} - k_j} \right) d_{\ell}^{k_{\ell - 1}} . \] This bound can be seen as a weak extension of the classical Bezout inequality (which holds only over algebraically closed fields and is provably false over real closed fields) to varieties defined over real closed fields. Additionally, if $\mathcal{P} \subset \mathrm{R}[X_1, \ldots, X_k] \nocomma$ is a finite family of polynomials with $\deg (P) \leq d$ for all $P \in \mathcal{P}$, $\mathrm{card} \mathcal{P}= s$, and $d_{\ell} \leq \frac{1}{k + 1} d_{}$, we prove that the number of semi-algebraically connected components of the realizations of all realizable sign conditions of the family $\mathcal{P}$ restricted to $V_{\ell}$ is bounded by \[ O (k)^{2 k} (s d)^{k_{\ell}} \left(\prod_{1 \leq j \leq \ell} d_j^{k_{j - 1} - k_j} \right) .