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

TL;DR

This paper demonstrates that while pathological zero set behaviors of polynomials on definable hypersurfaces exist, they are rare in a measure-theoretic sense, contrasting with the algebraic case.

Contribution

It constructs examples of definable hypersurfaces with arbitrarily complex polynomial zero sets and quantifies their rarity using measure bounds.

Findings

Pathological examples exist but have small measure.

Betti numbers of intersections can grow arbitrarily large.

Measure bounds depend on polynomial degree and hypersurface.

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 $\Gamma\subset \mathbb{R}\mathrm{P}^n$ 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.