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Smooth rigidity and Remez inequalities via Topology of level sets 基于水平集拓扑的光滑刚性和Remez不等式
IF 0.4
Journal of Singularities Pub Date : 2021-06-13 DOI: 10.5427/jsing.2022.25v
Y. Yomdin
{"title":"Smooth rigidity and Remez inequalities via Topology of level sets","authors":"Y. Yomdin","doi":"10.5427/jsing.2022.25v","DOIUrl":"https://doi.org/10.5427/jsing.2022.25v","url":null,"abstract":"A smooth rigidity inequalitiy provides an explicit lower bound for the (d+1)st derivatives of a smooth function f , which holds, if f exhibits certain patterns, forbidden for polynomials of degree d. The main goal of the present paper is twofold: first, we provide an overview of some recent results and questions related to smooth rigidity, which recently were obtained in Singularity Theory, in Approximation Theory, and in Whitney smooth extensions. Second, we prove some new results, specifically, a new Remez-type inequality, and on this base we obtain a new rigidity inequality. In both parts of the paper we stress the topology of the level sets, as the input information. Here are the main new results of the paper: Let B be the unit n-dimensional ball. For a given integer d let Z ⊂ B be a smooth compact hypersurface with N = (d − 1) + 1 connected components Zj . Let μj be the n-volume of the interior of Zj, and put μ = minμj , j = 1, . . . , N . Then for each polynomial P of degree d on R we have maxBn |P | max Z |P | ≤ ( 4n μ ). As a consequence, we provide an explicit lower bound for the (d+1)-st derivatives of any smooth function f , which vanishes on Z, while being of order 1 on B (smooth rigidity): ||f || ≥ 1 (d+ 1)! ( 4n μ ). We also provide an interpretation, in terms of smooth rigidity, of one of the simplest versions of the results in [8].","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77938837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A generalization of Zakalyukin's lemma, and symmetries of surface singularities Zakalyukin引理的推广及曲面奇点的对称性
IF 0.4
Journal of Singularities Pub Date : 2021-04-08 DOI: 10.5427/jsing.2022.25m
Atsufumi Honda, K. Naokawa, K. Saji, M. Umehara, Kotaro Yamada
{"title":"A generalization of Zakalyukin's lemma, and symmetries of surface singularities","authors":"Atsufumi Honda, K. Naokawa, K. Saji, M. Umehara, Kotaro Yamada","doi":"10.5427/jsing.2022.25m","DOIUrl":"https://doi.org/10.5427/jsing.2022.25m","url":null,"abstract":"Zakalyukin’s lemma asserts that the coincidence of the images of two wave front germs implies the right equivalence of corresponding map germs under a certain genericity assumption. The purpose of this paper is to give an improvement of this lemma for frontals. Moreover, we give several applications for singularities on surfaces.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76703340","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Algebraic differential equations of period-integrals 周期积分的代数微分方程
IF 0.4
Journal of Singularities Pub Date : 2021-01-25 DOI: 10.5427/jsing.2022.25c
D. Barlet
{"title":"Algebraic differential equations of period-integrals","authors":"D. Barlet","doi":"10.5427/jsing.2022.25c","DOIUrl":"https://doi.org/10.5427/jsing.2022.25c","url":null,"abstract":"We explain that in the study of the asymptotic expansion at the origin of a period integral like ∫ γz ω/df or of a hermitian period like ∫ f=s ρ.ω/df ∧ ω′/df the computation of the Bernstein polynomial of the ”fresco” (filtered differential equation) associated to the pair of germs (f, ω) gives a better control than the computation of the Bernstein polynomial of the full Brieskorn module of the germ of f at the origin. Moreover, it is easier to compute as it has a better functoriality and smaller degree. We illustrate this in the case where f ∈ C[x0, . . . , xn] has n+ 2 monomials and is not quasi-homogeneous, by giving an explicite simple algorithm to produce a multiple of the Bernstein polynomial when ω is a monomial holomorphic volume form. Several concrete examples are given. AMS Classification. 32 S 2532 S 40","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74621931","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets 紧亚解析集中正规电流和积分电流的线性等周不等式
IF 0.4
Journal of Singularities Pub Date : 2020-12-04 DOI: 10.5427/jsing.2022.24f
T. Pauw, R. Hardt
{"title":"Linear isoperimetric inequality for normal and integral currents in compact subanalytic sets","authors":"T. Pauw, R. Hardt","doi":"10.5427/jsing.2022.24f","DOIUrl":"https://doi.org/10.5427/jsing.2022.24f","url":null,"abstract":"The isoperimetric inequality for a smooth compact Riemannian manifold $A$ provides a positive ${bf c}(A)$, so that for any $k+1$ dimensional integral current $S_0$ in $A$ there exists an integral current $ S$ in $A$ with $partial S=partial S_0$ and ${bf M}(S)leq {bf c}(A){bf M}(partial S)^{(k+1)/k}$. Although such an inequality still holds for any compact Lipschitz neighborhood retract $A$, it may fail in case $A$ contains a single polynomial singularity. Here, replacing $(k+1)/k$ by $1$, we find that a linear inequality ${bf M}(S)leq {bf c}(A){bf M}(partial S)$ is valid for any compact algebraic, semi-algebraic, or even subanalytic set $A$. In such a set, this linear inequality holds not only for integral currents, which have $boldsymbol{Z}$ coefficients, but also for normal currents having $boldsymbol{R}$ coefficients and generally for normal flat chains with coefficients in any complete normed abelian group. A relative version for a subanalytic pair $Bsubset A$ is also true, and there are applications to variational and metric properties of subanalytic sets.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88302835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
Derived KZ Equations 导出的KZ方程
IF 0.4
Journal of Singularities Pub Date : 2020-12-04 DOI: 10.5427/jsing.2022.25u
V. Schechtman, A. Varchenko
{"title":"Derived KZ Equations","authors":"V. Schechtman, A. Varchenko","doi":"10.5427/jsing.2022.25u","DOIUrl":"https://doi.org/10.5427/jsing.2022.25u","url":null,"abstract":"In this paper we strengthen the results of [SV] by presenting their derived version. Namely, we define a \"derived Knizhnik - Zamolodchikov connection\" and identify it with a \"derived Gauss - Manin connection\".","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83060690","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties 齐次本质分离行列式变种的特征类
IF 0.4
Journal of Singularities Pub Date : 2020-11-25 DOI: 10.5427/jsing.2022.25w
Xiping Zhang
{"title":"Characteristic Classes of Homogeneous Essential Isolated Determinantal Varieties","authors":"Xiping Zhang","doi":"10.5427/jsing.2022.25w","DOIUrl":"https://doi.org/10.5427/jsing.2022.25w","url":null,"abstract":"The (homogeneous) Essentially Isolated Determinantal Variety is the natural generalization of generic determinantal variety, and is fundamental example to study non-isolated singularities. In this paper we study the characteristic classes on these varieties. We give explicit formulas of their Chern-Schwartz-MacPherson classes via standard Schubert calculus. As corollaries we obtain formulas for their (generic) sectional Euler characteristics, characteristic cycles and polar classes. In particular, when such variety is a hypersurfaces we compute its Milnor class and the Euler characteristics of the local Milnor fibers. We prove that for such recursive group orbit hypersurfaces the local Euler obstructions completely determine the Milnor classes. \u0000In general for reflective group orbits, on the other hand we propose an algorithm to compute their local Euler obstructions via the Chern-Schwartz-MacPherson classes of the orbits, which can be obtained directly from representation theory. This builds a bridge from representation theory of the group action to the singularity theory of the induced orbits.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72400793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Orbifold splice quotients and log covers of surface pairs 曲面对的轨道拼接商和对数覆盖
IF 0.4
Journal of Singularities Pub Date : 2020-11-18 DOI: 10.5427/jsing.2021.23i
W. Neumann, J. Wahl
{"title":"Orbifold splice quotients and log covers of surface pairs","authors":"W. Neumann, J. Wahl","doi":"10.5427/jsing.2021.23i","DOIUrl":"https://doi.org/10.5427/jsing.2021.23i","url":null,"abstract":"A three-dimensional orbifold $(Sigma, gamma_i, n_i)$, where $Sigma$ is a rational homology sphere, has a universal abelian orbifold covering, whose covering group is the first orbifold homology. A singular pair $(X,C)$, where $X$ is a normal surface singularity with $mathbb Q$HS link and $C$ is a Weil divisor, gives rise on its boundary to an orbifold. One studies the preceding orbifold notions in the algebro-geometric setting, in particular defining the universal abelian log cover of a pair. A first key theorem computes the orbifold homology from an appropriate resolution of the pair. In analogy with the case where $C$ is empty and one considers the universal abelian cover, under certain conditions on a resolution graph one can construct pairs and their universal abelian log covers. Such pairs are called orbifold splice quotients.","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90755791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
What is the degree of a smooth hypersurface? 光滑超曲面的度是多少?
IF 0.4
Journal of Singularities Pub Date : 2020-10-27 DOI: 10.5427/jsing.2021.23l
A. Lerário, Michele Stecconi
{"title":"What is the degree of a smooth hypersurface?","authors":"A. Lerário, Michele Stecconi","doi":"10.5427/jsing.2021.23l","DOIUrl":"https://doi.org/10.5427/jsing.2021.23l","url":null,"abstract":"Let $D$ be a disk in $mathbb{R}^n$ and $fin C^{r+2}(D, mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of $f$ meets a given semialgebraic set $Wsubset J^{r}(D, mathbb{R}^k).$ Examples of sets arising in this way are the zero set of $f$, or the set of its critical points. \u0000Under some transversality conditions, we prove that $f$ can be approximated with a polynomial map $p:Dto mathbb{R}^k$ such that the corresponding singularity is diffeomorphic to the original one, and such that the degree of this polynomial map can be controlled by the $C^{r+2}$ data of $f$. More precisely, begin{equation} text{deg}(p)le Oleft(frac{|f|_{C^{r+2}(D, mathbb{R}^k)}}{mathrm{dist}_{C^{r+1}}(f, Delta_W)}right), end{equation} \u0000where $Delta_W$ is the set of maps whose $r$-th jet extension is not transverse to $W$. The estimate on the degree of $p$ implies an estimate on the Betti numbers of the singularity, however, using more refined tools, we prove independently a similar estimate, but involving only the $C^{r+1}$ data of $f$. \u0000These results specialize to the case of zero sets of $fin C^{2}(D, mathbb{R})$, and give a way to approximate a smooth hypersurface defined by the equation $f=0$ with an algebraic one, with controlled degree (from which the title of the paper). In particular, we show that a compact hypersurface $Zsubset Dsubset mathbb{R}^n$ with positive reach $rho(Z)>0$ is isotopic to the zero set in $D$ of a polynomial $p$ of degree begin{equation} text{deg}(p)leq c(D)cdot 2 left(1+frac{1}{rho(Z)}+frac{5n}{rho(Z)^2}right),end{equation} where $c(D)>0$ is a constant depending on the size of the disk $D$ (and in particular on the diameter of $Z$).","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88342269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 3
On families of Lagrangian submanifolds 关于拉格朗日子流形的族
IF 0.4
Journal of Singularities Pub Date : 2020-10-22 DOI: 10.5427/jsing.2020.21j
S. Izumiya, Masatomo Takahashi
{"title":"On families of Lagrangian submanifolds","authors":"S. Izumiya, Masatomo Takahashi","doi":"10.5427/jsing.2020.21j","DOIUrl":"https://doi.org/10.5427/jsing.2020.21j","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84195655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the growth behaviour of Hironaka quotients 关于Hironaka商的增长行为
IF 0.4
Journal of Singularities Pub Date : 2020-10-22 DOI: 10.5427/jsing.2020.21o
Junki Tanaka, T. Ohmoto
{"title":"On the growth behaviour of Hironaka quotients","authors":"Junki Tanaka, T. Ohmoto","doi":"10.5427/jsing.2020.21o","DOIUrl":"https://doi.org/10.5427/jsing.2020.21o","url":null,"abstract":"","PeriodicalId":44411,"journal":{"name":"Journal of Singularities","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2020-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88162881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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