  Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact
hypersurfaces in $\mathbb{R}^{n-1}$, we prove that (up to extracting
subsequences) there exists a regular definable hypersurface $\Gamma\subset
\mathbb{R}\mathrm{P}^n$ such that each manifold $Z_d$ appears as a component of
the zero set on $\Gamma$ of some polynomial of degree $d$. (This is in sharp
contrast with the case when $\Gamma$ is algebraic, where for example the
homological complexity of the zero set of a polynomial $p$ on $\Gamma$ is
bounded by a polynomial in $\mathrm{deg}(p)$.) We call these "pathological
examples".
  In particular, we show that for every $0 \leq k \leq n-2$ and every sequence
of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact
and definable hypersurface $\Gamma\subset \mathbb{R}\mathrm{P}^n$, a
subsequence $\{a_{d_m}\}_{m\in \mathbb{N}}$ and homogeneous polynomials
$\{p_{m}\}_{m\in \mathbb{N}}$ of degree $\mathrm{deg}(p_m)=d_m$ such that:
\begin{equation} \label{eq:pathintro} b_k(\Gamma\cap Z(p_m))\geq
a_{d_m}.\end{equation} (Here $b_k$ denotes the $k$-th Betti number.) This
generalizes a result of Gwo\'zdziewicz, Kurdyka and Parusi\'nski.
  On the other hand, for a given definable $\Gamma$ we show that the
Fubini-Study measure, in the gaussian space of polynomials of degree $d$, of
the set $\Sigma_{d_m,a, \Gamma}$ of polynomials verifying $b_k(\Gamma\cap
Z(p_m))\geq a_{d_m}$ is positive, but there exists a contant $c_\Gamma$ such
that this measure can be bounded by: \begin{equation} 0<\mathbb{P}(\Sigma_{d_m,
a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}. \end{equation}
This shows that the set of "pathological examples" has "small" measure.
