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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   $\Gamma \subset {\mathbb{R}}\textrm{P}^n$   such that each manifold   $Z_d$   is diffeomorphic to 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 semialgebraic, where for example the homological complexity of the zero set of a polynomial   $p$   on   $\Gamma$   is bounded by a polynomial in   $\deg (p)$  .) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface   $\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$   containing a subset   $D$   homeomorphic to a disk, and a family of polynomials   $\{p_m\}_{m\in \mathbb{N}}$   of degree   $\deg (p_m)=d_m$   such that   $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$   i.e. the zero set of   $p_m$   in   $D$   is isotopic to   $Z_{d_m}$   in   ${\mathbb{R}}^{n-1}$  . This says that, up to extracting subsequences, the intersection of   $\Gamma$   with a hypersurface of degree   $d$   can be as complicated as we want. 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 semianalytic hypersurface   $\Gamma \subset {\mathbb{R}}\textrm{P}^n$  , a subsequence   $\{a_{d_m}\}_{m\in \mathbb{N}}$   and homogeneous polynomials   $\{p_{m}\}_{m\in \mathbb{N}}$   of degree   $\deg (p_m)=d_m$   such that   (0.1)  $$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$   (Here   $b_k$   denotes the   $k$  th Betti number.) This generalizes a result of Gwoździewicz  et al . [ 13 ]. On the other hand, for a given definable   $\Gamma$   we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree   $d$  , of the set   $\Sigma _{d_m,a, \Gamma }$   of polynomials verifying ( 0.1 ) is positive, but there exists a constant   $c_\Gamma$   such that   $$\begin{equation*}0&lt;{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$   This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster   $a$   grows, the smaller the measure and pathologies are therefore rare). In fact we show that given   $\Gamma$  , for most polynomials a Bézout-type bound holds for the intersection   $\Gamma \cap Z(p)$  : for every   $0\leq k\leq n-2$   and   $t&gt;0$  :   $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$

Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare