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.
Categories: math.AG