Let $V$ be a a real variety in $\mathbb{R}^n$, defined by the solution set of a real polynomial system $f_i(x_1,x_2,...,x_n)=0, i=1,2,...,m$.
#1. We can say the number of connected components of $V$ is bounded by a number. Like theorem 5.1 says here.
#2. I also found a paper claims there is a bound for number of real solutions, where the assumption is that exponent vectors of the polynomial systems generate a subgroup of $\mathbb{Z}^n$ of odd index. (What dose this mean? I can't understand this. Does this ensure the bounded number of solutions? ).
In my understanding, a real polynomial system can have infinite number of real solutions, but may have limited number of connected components. How are these two bounds (#1 and #2) related? Or generally, can we say something about the relationship between the number of roots and the number of connected components?