Suppose an algebraic variety $V$ is given as the solutions to $q$ polynomial equations of degree $\le k$ with real coefficients $$p_1(x_1,\dots,x_m)=0,\dots,p_q(x_1,\dots,x_m)=0$$ for $x\in\mathbb R^m$. It is a theorem due to Milnor that the sum of Betti numbers of $V$ is bounded by $k(2k-1)^{m-1}$. In particular this also bounds the number of connected components of $V$.

**Question:** Is this bound on the number of connected components of $V$ the best known bound? I am only interested in the case $k=2$. I suspect that for the varieties I am studying the number of connected components is bounded linearly in $m$.

*Milnor, John W.*, **On the Betti numbers of real varieties**, Proc. Am. Math. Soc. 15, 275-280 (1964). ZBL0123.38302.