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]$suchthatforeach$i, 1 \leq i \leq \ell$,$$\mathrm{d}\mathrm{e}\mathrm{g}$ (Q_i) \leq d_i$\mathrm{.}For$1 \leq i \leq \ell$,denoteby$$\mathcal{Q}$_i = {Q_1, \ldots, Q_i }$,$V_i$therealvarietydefinedby$$\mathcal{Q}$i$,and$k_i$anupperboundontherealdimensionof$V_i$ (by convention $V_0={\mathrm{R}}^k$ and $k_0=k$). Suppose also that $$2\le d_1\le d_2\le \frac{1}{k+1}{d}_3\le \frac{1}{(k+1{)}^2}{d}_4\le \cdots \le \frac{1}{(k+1{)}^{\mathrm{\ell }-3}}{d}_{\mathrm{\ell }-1}\le \frac{1}{(k+1{)}^{\mathrm{\ell }-2}}{d}_{\mathrm{\ell }},$$ and that $\mathrm{\ell }\le k$. We prove that the number of semi-algebraically connected components of $V_{\mathrm{\ell }}$ is bounded by $$O(k{)}^{2k}\left(\prod _{1\le j<\mathrm{\ell }}{d}_j^{k_{j-1}-k_j}\right)d_{\mathrm{\ell }}^{k_{\mathrm{\ell }-1}}\mathrm{.}$$ This bound can be seen as a weak extension of the classical Bezout inequality (which holds only over algebraically closed fields and is false over real closed fields) to varieties defined over real closed fields. Additionally, if $\mathcal{P}\subset \mathrm{R}[X_1,\dots ,X_k]$ is a finite family of polynomials with $\mathrm{d}\mathrm{e}\mathrm{g}(P)\le d$ for all $P \in $\mathcal{P}$$,$$\mathrm{c}\mathrm{a}\mathrm{r}\mathrm{d}$( $\mathcal{P}$) = 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 $V_{\mathrm{\ell }}$ is bounded by $$O(k{)}^{2k}(sd{)}^{k_{\mathrm{\ell }}}\left(\prod _{1\le j\le \mathrm{\ell }}{d}_j^{k_{j-1}-k_j}\right)\mathrm{.}$$