On a real analogue of Bezout inequality and the number 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 $i, 1 \leq i \leq \ell$, $\mathrm{deg} (Q_i) \leq d_i$. For $1 \leq i \leq \ell$, denote by $\mathcal{Q}_i = \{Q_1, \ldots, Q_i \}$, $V_i$ the real variety defined by $\mathcal{Q}_i$, and $k_i$ an upper bound on the real dimension of $V_i$ (by convention $V_0 = \mathrm{R}^k$ 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 false over real closed fields) to varieties defined over real closed fields. Additionally, if $\mathcal{P} \subset \mathrm{R}[X_1, \ldots, X_k]$ is a finite family of polynomials with $\mathrm{deg} (P) \leq d$ 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 \[ O (k)^{2 k} (s d)^{k_{\ell}} \left(\prod_{1 \leq j \leq \ell} d_j^{k_{j - 1} - k_j} \right). \]