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].
光滑刚性不等式为光滑函数f的(d+1)st阶导数提供了一个显式下界，如果f表现出某些模式，则对d次多项式是禁止的。本文的主要目标有两个:首先，我们概述了最近在奇点理论、近似理论和惠特尼光滑扩展中获得的与光滑刚性有关的一些结果和问题。其次，我们证明了一些新的结果，特别是一个新的remez型不等式，并在此基础上得到了一个新的刚性不等式。在本文的两部分中，我们都强调水平集的拓扑结构作为输入信息。以下是本文的主要新结果:设B为单位n维球。对于给定的整数d，设Z∧B是一个光滑紧超曲面，具有N = (d−1)+ 1个连通分量Zj。设μj为Zj内部的n体积，设μ = minμj, j = 1，…。，名词;那么对于R上的每个d次多项式P，我们有maxBn |P | max Z |P |≤(4n μ)。因此，我们为任意光滑函数f的(d+1)-st阶导数提供了一个显式下界，该函数在Z上消失，而在B上是1阶(光滑刚性):||f ||≥1 (d+1) !(4n μ)。我们还对[8]中结果的一个最简单版本提供了平滑刚性的解释。
{"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}
Atsufumi Honda, K. Naokawa, K. Saji, M. Umehara, Kotaro Yamada
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.
{"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}
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
{"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}
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.
{"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}
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".
{"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}
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. In 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.
{"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}
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.
{"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}
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. Under 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} where $Delta_W$ 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 $f$. These 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 $Z$).
{"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}
{"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}
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