Let $n$ and $d$ be positive integers, and $f\in\mathbb{R}[x_1,\dots,x_n]$ be a polynomial of degree $d$. Let's consider the zero-set $M = \{x \in \mathbb{R}^n: f(x) = 0\}$ of $f$.

Can we estimate the number of connected components of $\mathbb{R}^n\setminus M$?

I throw out a guess: no more than $2^d$. An realisation of the bound $2^d$ is given by the example: $f(x) = x_1x_2\cdots x_d$ (when $n \ge d$). I can prove it for the case $d = 2$.

UPDATE: is it possible to find an estimate that does not depend on $n$?