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

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}$^n$suchthateachmanifold$Z_d$ appears as a component of the zero set on $\mathrm{\Gamma }$ of some polynomial of degree $d$. (This is in sharp contrast with the case when $\mathrm{\Gamma }$ is algebraic, where for example the homological complexity of the zero set of a polynomial $p$ on $\mathrm{\Gamma }$ is bounded by a polynomial in $\mathrm{d}\mathrm{e}\mathrm{g}(p)$.) We call these “pathological examples”. In particular, we show that for every $0\le k\le n-2$ and every sequence of natural numbers $a=\{a_d{\}}_{d\in \mathbb{N}}$ there is a regular, compact and definable hypersurface $\mathrm{\Gamma }\subset \mathbb{R}{\mathrm{P}}^n$, a subsequence $\{a_{d_m}{\}}_{m\in \mathbb{N}}$ and homogeneous polynomials $\{p_m{\}}_{m\in \mathbb{N}}$ of degree $\mathrm{d}\mathrm{e}\mathrm{g}(p_m)=d_m$ 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 $\mathrm{\Gamma }$ we show that the Fubini-Study measure, in the gaussian space of polynomials of degree $d$, of the set ${\mathrm{\Sigma }}_{d_m,a,\mathrm{\Gamma }}$ of polynomials verifying $b_k(\Gamma\cap Z(p_m))\geq a_{d_m}$ispositive,butthereexistsacontant$c_\Gamma$ 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.