Pub Date : 2024-06-28DOI: 10.1007/s40687-024-00459-6
Federica Galluzzi, Bert van Geemen
A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the K3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.
{"title":"Invariants of vanishing Brauer classes","authors":"Federica Galluzzi, Bert van Geemen","doi":"10.1007/s40687-024-00459-6","DOIUrl":"https://doi.org/10.1007/s40687-024-00459-6","url":null,"abstract":"<p>A specialization of a <i>K</i>3 surface with Picard rank one to a <i>K</i>3 with rank two defines a vanishing class of order two in the Brauer group of the general <i>K</i>3 surface. We give the <i>B</i>-field invariants of this class. We apply this to the <i>K</i>3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the <i>K</i>3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-26DOI: 10.1007/s40687-024-00458-7
C. Bivià-Ausina, K. Kourliouros, M. A. S. Ruas
Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.
给定一个解析变种 X 的胚芽和一个全纯函数 f 的胚芽,该函数 f 相对于 X 的对数分层具有分层孤立奇点,我们证明,在一对(f, X)的奇点类型的某些条件下,著名的 K. Saito 定理的以下相对类似定理成立:f 相对于 X 的相对米尔诺数和特尤里纳数(也称为布鲁斯-罗伯茨数)相等,等同于相对准均质性。Saito 定理的以下相对类比定理成立:f 相对于 X 的相对 Milnor 数和 Tjurina 数(也称为 Bruce-Roberts 数)的相等等价于一对(f, X)的相对准均质性,即存在一个坐标系,使得 f 和 X 相对于相同的正有理权重都是准均质的。
{"title":"Bruce–Roberts numbers and quasihomogeneous functions on analytic varieties","authors":"C. Bivià-Ausina, K. Kourliouros, M. A. S. Ruas","doi":"10.1007/s40687-024-00458-7","DOIUrl":"https://doi.org/10.1007/s40687-024-00458-7","url":null,"abstract":"<p>Given a germ of an analytic variety <i>X</i> and a germ of a holomorphic function <i>f</i> with a stratified isolated singularity with respect to the logarithmic stratification of <i>X</i>, we show that under certain conditions on the singularity type of the pair (<i>f</i>, <i>X</i>), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of <i>f</i> with respect to <i>X</i> (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (<i>f</i>, <i>X</i>), i.e. to the existence of a coordinate system such that both <i>f</i> and <i>X</i> are quasihomogeneous with respect to the same positive rational weights.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-26DOI: 10.1007/s40687-024-00457-8
José Ignacio Cogolludo-Agustín, Tamás László, Jorge Martín-Morales, András Némethi
In this article we study the delta invariant of reduced curve germs via topological techniques. We describe an explicit connection between the delta invariant of a curve embedded in a rational singularity and the topological Poincaré series of the ambient surface. This connection is established by using another formula expressing the delta invariant as ‘periodic constants’ of the Poincaré series associated with the abstract curve and a ‘twisted’ duality developed for the Poincaré series of the ambient space.
{"title":"Duality for Poincaré series of surfaces and delta invariant of curves","authors":"José Ignacio Cogolludo-Agustín, Tamás László, Jorge Martín-Morales, András Némethi","doi":"10.1007/s40687-024-00457-8","DOIUrl":"https://doi.org/10.1007/s40687-024-00457-8","url":null,"abstract":"<p>In this article we study the delta invariant of reduced curve germs via topological techniques. We describe an explicit connection between the delta invariant of a curve embedded in a rational singularity and the topological Poincaré series of the ambient surface. This connection is established by using another formula expressing the delta invariant as ‘periodic constants’ of the Poincaré series associated with the abstract curve and a ‘twisted’ duality developed for the Poincaré series of the ambient space.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-24DOI: 10.1007/s40687-024-00456-9
Najib Idrissi, Eugene Rabinovich
We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space M. This allows us to describe a notion of prefactorization algebra up to homotopy as well as morphisms up to homotopy between such objects. We make explicit these notions for several special M, such as certain finite topological spaces, or the real line.
我们应用操作数科斯祖尔对偶性理论,提供了彩色操作数的共纤解析,其代数是固定空间 M 上的前因式分解代数。这使我们能够描述前因式分解代数直到同调的概念,以及这些对象之间直到同调的态量。我们明确了几个特殊 M 的这些概念,如某些有限拓扑空间或实线。
{"title":"Homotopy prefactorization algebras","authors":"Najib Idrissi, Eugene Rabinovich","doi":"10.1007/s40687-024-00456-9","DOIUrl":"https://doi.org/10.1007/s40687-024-00456-9","url":null,"abstract":"<p>We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space <i>M</i>. This allows us to describe a notion of prefactorization algebra up to homotopy as well as morphisms up to homotopy between such objects. We make explicit these notions for several special <i>M</i>, such as certain finite topological spaces, or the real line.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141501530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-15DOI: 10.1007/s40687-024-00455-w
James Branch, Nikolaos Diamantis, W. Raji, Larry Rolen
{"title":"Period-like polynomials for L-series associated with half-integral weight cusp forms","authors":"James Branch, Nikolaos Diamantis, W. Raji, Larry Rolen","doi":"10.1007/s40687-024-00455-w","DOIUrl":"https://doi.org/10.1007/s40687-024-00455-w","url":null,"abstract":"","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141336653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-18DOI: 10.1007/s40687-024-00450-1
A. C. Nabarro, M. C. Romero Fuster, M. C. Zanardo
The parabolic subset of a 3-manifold generically immersed in (mathbb {R}^4) is a surface. We analyze in this study the generic geometrical behavior of such surface, considered as a submanifold of (mathbb {R}^4). Typical Singularity Theory techniques based on the analysis of the family of height functions are applied in order to describe the geometrical characterizations of different singularity types.
{"title":"Geometry of the parabolic subset of generically immersed 3-manifolds in $$mathbb {R}^4$$","authors":"A. C. Nabarro, M. C. Romero Fuster, M. C. Zanardo","doi":"10.1007/s40687-024-00450-1","DOIUrl":"https://doi.org/10.1007/s40687-024-00450-1","url":null,"abstract":"<p>The parabolic subset of a 3-manifold generically immersed in <span>(mathbb {R}^4)</span> is a surface. We analyze in this study the generic geometrical behavior of such surface, considered as a submanifold of <span>(mathbb {R}^4)</span>. Typical Singularity Theory techniques based on the analysis of the family of height functions are applied in order to describe the geometrical characterizations of different singularity types.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141061752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-11DOI: 10.1007/s40687-024-00447-w
Amanda Folsom, David Metacarpa
Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of “universal quantum mock theta function,” in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an “antiquantum q-series identity’ and offer motivating general results with applications to third-order mock theta functions.
{"title":"Quantum q-series and mock theta functions","authors":"Amanda Folsom, David Metacarpa","doi":"10.1007/s40687-024-00447-w","DOIUrl":"https://doi.org/10.1007/s40687-024-00447-w","url":null,"abstract":"<p>Our results investigate mock theta functions and quantum modular forms via quantum <i>q</i>-series identities. After Lovejoy, quantum <i>q</i>-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum <i>q</i>-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of “universal quantum mock theta function,” in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum <i>q</i>-series identity and the results of this paper, we also define what we call an “antiquantum <i>q</i>-series identity’ and offer motivating general results with applications to third-order mock theta functions.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941901","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-05DOI: 10.1007/s40687-024-00453-y
Maico Ribeiro, Ivan Santamaria, Thiago da Silva
In this paper, we discuss the concept of (rho )-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function (rho ), which guarantees (rho )-regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.
{"title":"Some remarks about $$ rho $$ -regularity for real analytic maps","authors":"Maico Ribeiro, Ivan Santamaria, Thiago da Silva","doi":"10.1007/s40687-024-00453-y","DOIUrl":"https://doi.org/10.1007/s40687-024-00453-y","url":null,"abstract":"<p>In this paper, we discuss the concept of <span>(rho )</span>-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map <i>G</i> with respect to the levels of a function <span>(rho )</span>, which guarantees <span>(rho )</span>-regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937465","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-02DOI: 10.1007/s40687-024-00452-z
Nicholas Anderson
Symmetric powers of matroids were first introduced by Lovasz (Combinatorial surveys, in: Proceedings 6th British combinatorial conference, pp 45-86, 1977) and Mason (Algebr Methods Graph Theory 1:519-561, 1981) in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor-closed and has infinitely many forbidden minors.
{"title":"Matroid products in tropical geometry","authors":"Nicholas Anderson","doi":"10.1007/s40687-024-00452-z","DOIUrl":"https://doi.org/10.1007/s40687-024-00452-z","url":null,"abstract":"<p>Symmetric powers of matroids were first introduced by Lovasz (Combinatorial surveys, in: Proceedings 6th British combinatorial conference, pp 45-86, 1977) and Mason (Algebr Methods Graph Theory 1:519-561, 1981) in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor-closed and has infinitely many forbidden minors.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-02DOI: 10.1007/s40687-024-00448-9
G. Goldman, Y. Yomdin
Let (f: B^n rightarrow {{mathbb {R}}}) be a (d+1) times continuously differentiable function on the unit ball (B^n), with (mathrm{max,}_{zin B^n} Vert f(z) Vert =1). A well-known fact is that if f vanishes on a set (Zsubset B^n) with a non-empty interior, then for each (k=1,ldots ,d+1) the norm of the k-th derivative (||f^{(k)}||) is at least (M=M(n,k)>0). A natural question to ask is “what happens for other sets Z?”. This question was partially answered in Goldman and Yomdin (Lower bounds for high derivatives of smooth functions with given zeros. arXiv:2402.01388), Yomdin (Anal Math Phys 11:89, 2021), Yomdin (J Singul 25:443–455, 2022) and Yomdin (Higher derivatives of functions vanishing on a given set. arXiv:2108.02459v1). In the present paper, we ask a similar (and closely related) question: what happens with the high-order derivatives of f, if its gradient vanishes on a given set (Sigma )? And what conclusions for the high-order derivatives of f can be obtained from the analysis of the metric geometry of the “critical values set” (f(Sigma ))? In the present paper, we provide some initial answers to these questions.
让(f: B^n 是单位球上的(d+1)次连续可微分函数,其中({mathrm{max,}_{zin B^n}f(z) =1)。一个众所周知的事实是,如果f在一个具有非空内部的集合(Z子集B^n)上消失,那么对于每一个(k=1,dots ,d+1),k-导数(||f^{(k)}||||)的规范至少是(M=M(n,k)>0).一个自然的问题是 "其他集合 Z 会怎样?这个问题在 Goldman 和 Yomdin (Lower bounds for high derivatives of smooth functions with given zeros. arXiv:2402.01388), Yomdin (Anal Math Phys 11:89, 2021), Yomdin (J Singul 25:443-455, 2022) 和 Yomdin (Higher derivatives of functions vanishing on a given set. arXiv:2108.02459v1) 中得到了部分回答。在本文中,我们提出了一个类似(且密切相关)的问题:如果 f 的梯度在给定集合 (Sigma ) 上消失,那么 f 的高阶导数会发生什么变化?通过对 "临界值集"(f(Sigma )的度量几何的分析,可以得到关于 f 的高阶导数的哪些结论?)在本文中,我们将为这些问题提供一些初步的答案。
{"title":"Higher derivatives of functions with given critical points and values","authors":"G. Goldman, Y. Yomdin","doi":"10.1007/s40687-024-00448-9","DOIUrl":"https://doi.org/10.1007/s40687-024-00448-9","url":null,"abstract":"<p>Let <span>(f: B^n rightarrow {{mathbb {R}}})</span> be a <span>(d+1)</span> times continuously differentiable function on the unit ball <span>(B^n)</span>, with <span>(mathrm{max,}_{zin B^n} Vert f(z) Vert =1)</span>. A well-known fact is that if <i>f</i> vanishes on a set <span>(Zsubset B^n)</span> with a non-empty interior, then for each <span>(k=1,ldots ,d+1)</span> the norm of the <i>k</i>-th derivative <span>(||f^{(k)}||)</span> is at least <span>(M=M(n,k)>0)</span>. A natural question to ask is “what happens for other sets <i>Z</i>?”. This question was partially answered in Goldman and Yomdin (Lower bounds for high derivatives of smooth functions with given zeros. arXiv:2402.01388), Yomdin (Anal Math Phys 11:89, 2021), Yomdin (J Singul 25:443–455, 2022) and Yomdin (Higher derivatives of functions vanishing on a given set. arXiv:2108.02459v1). In the present paper, we ask a similar (and closely related) question: what happens with the high-order derivatives of <i>f</i>, if its gradient vanishes on a given set <span>(Sigma )</span>? And what conclusions for the high-order derivatives of <i>f</i> can be obtained from the analysis of the metric geometry of the “critical values set” <span>(f(Sigma ))</span>? In the present paper, we provide some initial answers to these questions.</p>","PeriodicalId":48561,"journal":{"name":"Research in the Mathematical Sciences","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937301","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}