Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare
Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare
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可定义超曲面上多项式的零点：病理现象存在，但很少见
DOI：
10.1093/qmath/haz022
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发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Natarajan, Abhiram
中科院分区：
文献类型：
--
作者：
Basu, Saugata;Lerario, Antonio;Natarajan, Abhiram
Given a sequenceof smooth and compact hypersurfaces in, we prove that (up to extracting subsequences) there exists a regular definable hypersurfacesuch that each manifoldis diffeomorphic to a component of the zero set onof some polynomial of degree. (This is in sharp contrast with the case whenis semialgebraic, where for example the homological complexity of the zero set of a polynomialonis bounded by a polynomial in.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurfacecontaining a subsethomeomorphic to a disk, and a family of polynomialsof degreesuch thati.e. the zero set ofinis isotopic toin. This says that, up to extracting subsequences, the intersection ofwith a hypersurface of degreecan be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for everyand every sequence of natural numbersthere is a regular, compact semianalytic hypersurface, a subsequenceand homogeneous polynomialsof degreesuch that $$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Heredenotes theth Betti number.) This generalizes a result of Gwoździewiczet al. . On the other hand, for a given definablewe show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree, of the setof polynomials verifying is positive, but there exists a constantsuch 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 fastergrows, the smaller the measure and pathologies are therefore rare). In fact we show that given, for most polynomials a Bézout-type bound holds for the intersection: for everyand: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$
Given a sequenceof smooth and compact hypersurfaces in, we prove that (up to extracting subsequences) there exists a regular definable hypersurfacesuch that each manifoldis diffeomorphic to a component of the zero set onof some polynomial of degree. (This is in sharp contrast with the case whenis semialgebraic, where for example the homological complexity of the zero set of a polynomialonis bounded by a polynomial in.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurfacecontaining a subsethomeomorphic to a disk, and a family of polynomialsof degreesuch thati.e. the zero set ofinis isotopic toin. This says that, up to extracting subsequences, the intersection ofwith a hypersurface of degreecan be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for everyand every sequence of natural numbersthere is a regular, compact semianalytic hypersurface, a subsequenceand homogeneous polynomialsof degreesuch that $$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Heredenotes theth Betti number.) This generalizes a result of Gwoździewiczet al. . On the other hand, for a given definablewe show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree, of the setof polynomials verifying is positive, but there exists a constantsuch 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 fastergrows, the smaller the measure and pathologies are therefore rare). In fact we show that given, for most polynomials a Bézout-type bound holds for the intersection: for everyand: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$
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影响因子：
1.5
作者：
Y. Fyodorov;A. Lerário;Erik Lundberg
通讯作者：
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DOI：
--
发表时间：
1999
期刊：
影响因子：
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作者：
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DOI：
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期刊：
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作者：
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DOI：
10.1515/crelle-2018-0009
发表时间：
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期刊：
Journal für die reine und angewandte Mathematik (Crelles Journal)
影响因子：
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作者：
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