Abstract

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  
$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$
(0.1)
(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  
$$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$
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 |$t>0$|⁠:  
$$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$
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