  Let $\mathrm{R}$ be a real closed field. We prove that the number of
semi-algebraically connected components of a real hypersurface in
$\mathrm{R}^n$ defined by a multi-affine polynomial of degree $d$ is bounded by
$2^{d-1}$. This bound is sharp and is independent of $n$ (as opposed to the
classical bound of $d(2d -1)^{n-1}$ on the Betti numbers of hypersurfaces
defined by arbitrary polynomials of degree $d$ in $\mathrm{R}^n$ due to
Petrovski{\u\i} and Ole{\u\i}nik, Thom and Milnor). Moreover, we show there
exists $c > 1$, such that given a sequence $(B_n)_{n >0}$ where $B_n$ is a
closed ball in $\mathrm{R}^n$ of positive radious, there exist hypersurfaces
$(V_n)_{n_>0}$ defined by symmetric multi-affine polynomials of degree $4$,
such that $\sum_{i \leq 5} b_i(V_n \cap B_n) > c^n$, where $b_i(\cdot)$ denotes
the $i$-th Betti number with rational coeffcients. Finally, as an application
of the main result of the paper we verify a representational stability
conjecture due to Basu and Riener on the cohomology modules of symmetric real
algebraic sets for a new and much larger class of symmetric real algebraic sets
than known before.
