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 (0.10.1) 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 $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$

中文翻译：

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可定义的超曲面上的多项式零点：存在病态，但很少见

给定序列$ \ {Z_d \} _ {d \ in \ mathbb {N}} $中$ {\ mathbb {R}} ^ {n-1} $中的光滑和紧凑的超曲面，我们证明（直到提取子序列），存在一个规则的可定义超曲面$ \ Gamma \ subset {\ mathbb {R}} \ textrm {P} ^ n $，这样每个流形$ Z_d $都会微分变换为$ \ Gamma $上零集的一个分量$ d $的多项式。（这与$ \ Gamma $是半代数的情况形成鲜明对比，例如$ \ Gamma $上的多项式$ p $的零集的齐次复杂度受$ \ deg（p）的多项式限制$。）更精确地讲，给定以上超曲面序列，我们构造一个规则的，紧凑的，半解析超曲面$ \ Gamma \ subset {\ mathbb {R}} \ textrm {P} ^ {n} $包含子集$ D $对磁盘同胚 和一阶多项式$ \ {p_m \} _ {m \ in \ mathbb {N}} $的度数$ \ deg（p_m）= d_m $，使得$（D，Z（p_m）\ cap D）\ sim （{\ mathbb {R}} ^ {n-1}，Z_ {d_m}），$，即$ D $中$ p_m $的零集与$ {\ mathbb {R }} ^ {n-1} $。这就是说，直到提取子序列，$ \ Gamma $与度数为$ d $的超曲面的交点可能会像我们想要的那样复杂。我们称这些为“病理例子”。特别地，我们表明，对于每个$ 0 \ leq k \ leq n-2 $和每个自然数序列$ a = \ {a_d \} _ {d \ in \ mathbb {N}} $，都有一个规则的，紧凑的半解析超曲面$ \ Gamma \ subset {\ mathbb {R}} \ textrm {P} ^ n $，一个子序列$ \ {a_ {d_m} \} _ {m \ in \ mathbb {N}} $和齐次多项式$ \ {p_ {m} \} _ {m \ in \ mathbb {N}} $的度数$ \ deg（p_m）= d_m $，使得（0.1）$$ \ begin {equation} b_k（\ Gamma \ cap Z （p_m））\ geq a_ {d_m}。\ end {equation} $$（此处$ b_k $表示第k个Betti数字。）这概括了Gwoździewicz等人的结果。[13]。另一方面，对于给定的$ \ Gamma $，我们证明了在集合$ \ Sigma _ {d_m，a，\ Gamma}的度为$ d $的多项式的高斯概率空间中，Fubini-Study测度验证多项式（0.10.1）的$为正，但是存在一个常数$ c_ \ Gamma $，使得$$ \ begin {equation *} 0 <{\ mathbb {P}}（\ Sigma_ {d_m，a，\ Gamma}）\ leq \ frac {c _ {\ Gamma} d_m ^ {\ frac {n-1} {2}}} {a_ {d_m}}。\ end {equation *} $$这表明“病理学例子”的衡量标准较小（$ a $增长得越快，衡量标准越小，因此很少出现病理学）。实际上，我们证明了给定$ \ Gamma $，对于大多数多项式，交集$ \ Gamma \ cap Z（p）$的Bézout型界成立：每$ 0 \ leq k \ leq n-2 $和$ t> 0 $：