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An Elekes–Rónyai Theorem for Sets With Few Products 少乘积集合的 Elekes-Rónyai 定理
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-07 DOI: 10.1093/imrn/rnae087
Akshat Mudgal
{"title":"An Elekes–Rónyai Theorem for Sets With Few Products","authors":"Akshat Mudgal","doi":"10.1093/imrn/rnae087","DOIUrl":"https://doi.org/10.1093/imrn/rnae087","url":null,"abstract":"Given $n in mathbb{N}$, we call a polynomial $F in mathbb{C}[x_{1},dots ,x_{n}]$ degenerate if there exist $Pin mathbb{C}[y_{1}, dots , y_{n-1}]$ and monomials $m_{1}, dots , m_{n-1}$ with fractional exponents, such that $F = P(m_{1}, dots , m_{n-1})$. Our main result shows that whenever a polynomial $F$, with degree $d geq 1$, is non-degenerate, then for every finite, non-empty set $Asubset mathbb{C}$ such that $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A, dots, A)| gg |A|^{n} 2^{-O_{d,n}((log 2K)^{3 + o(1)})}. end{align*} $$This is sharp since for every degenerate $F$ and finite set $A subset mathbb{C}$ with $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A,dots,A)| ll K^{O_{F}(1)}|A|^{n-1}.end{align*} $$Our techniques rely on Freiman type inverse theorems and Schmidt’s subspace theorem.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140937469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group 海森堡群中具有一般增长的非局部方程的正则理论
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-02 DOI: 10.1093/imrn/rnae072
Yuzhou Fang, Chao Zhang
{"title":"Regularity Theory for Nonlocal Equations with General Growth in the Heisenberg Group","authors":"Yuzhou Fang, Chao Zhang","doi":"10.1093/imrn/rnae072","DOIUrl":"https://doi.org/10.1093/imrn/rnae072","url":null,"abstract":"We deal with a wide class of generalized nonlocal $p$-Laplace equations, so-called nonlocal $G$-Laplace equations, in the Heisenberg framework. Under natural hypotheses on the $N$-function $G$, we provide a unified approach to investigate in the spirit of De Giorgi-Nash-Moser theory, some local properties of weak solutions to such kind of problems, involving boundedness, Hölder continuity and Harnack inequality. To this end, an improved nonlocal Caccioppoli-type estimate as the main auxiliary ingredient is exploited several times.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Parabolic Analog of a Theorem of Beilinson and Schechtman 贝林森和谢赫特曼定理的抛物线类比
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-05-02 DOI: 10.1093/imrn/rnae085
Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth
{"title":"A Parabolic Analog of a Theorem of Beilinson and Schechtman","authors":"Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth","doi":"10.1093/imrn/rnae085","DOIUrl":"https://doi.org/10.1093/imrn/rnae085","url":null,"abstract":"For a simple, simply connected, complex group $G$, we prove an explicit formula to compute the Atiyah class of parabolic determinant of cohomology line bundle on the moduli space of parabolic $G$-bundles. This generalizes an earlier result of Beilinson-Schechtman.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The Irreducibility of the Spaces of Rational Curves on del Pezzo Manifolds 德尔佩佐积分上有理曲线空间的不可还原性
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-29 DOI: 10.1093/imrn/rnae080
Fumiya Okamura
{"title":"The Irreducibility of the Spaces of Rational Curves on del Pezzo Manifolds","authors":"Fumiya Okamura","doi":"10.1093/imrn/rnae080","DOIUrl":"https://doi.org/10.1093/imrn/rnae080","url":null,"abstract":"We prove the irreducibility of the spaces of rational curves on del Pezzo manifolds of Picard rank $1$ and dimension $n ge 4$ by analyzing the fibers of evaluation maps. As a corollary, we prove Geometric Manin’s Conjecture in these cases.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Geometric Approach to Polynomial and Rational Approximation 多项式和有理数逼近的几何方法
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-29 DOI: 10.1093/imrn/rnae082
Christopher J Bishop, Kirill Lazebnik
{"title":"A Geometric Approach to Polynomial and Rational Approximation","authors":"Christopher J Bishop, Kirill Lazebnik","doi":"10.1093/imrn/rnae082","DOIUrl":"https://doi.org/10.1093/imrn/rnae082","url":null,"abstract":"We strengthen the classical approximation theorems of Weierstrass, Runge, and Mergelyan by showing the polynomial and rational approximants can be taken to have a simple geometric structure. In particular, when approximating a function $f$ on a compact set $K$, the critical points of our approximants may be taken to lie in any given domain containing $K$, and all the critical values in any given neighborhood of the polynomially convex hull of $f(K)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140831495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Extremal Numbers and Sidorenko’s Conjecture 极值数和西多连科猜想
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-24 DOI: 10.1093/imrn/rnae071
David Conlon, Joonkyung Lee, Alexander Sidorenko
{"title":"Extremal Numbers and Sidorenko’s Conjecture","authors":"David Conlon, Joonkyung Lee, Alexander Sidorenko","doi":"10.1093/imrn/rnae071","DOIUrl":"https://doi.org/10.1093/imrn/rnae071","url":null,"abstract":"Sidorenko’s conjecture states that, for all bipartite graphs $H$, quasirandom graphs contain asymptotically the minimum number of copies of $H$ taken over all graphs with the same order and edge density. While still open for graphs, the analogous statement is known to be false for hypergraphs. We show that there is some advantage in this, in that if Sidorenko’s conjecture does not hold for a particular $r$-partite $r$-uniform hypergraph $H$, then it is possible to improve the standard lower bound, coming from the probabilistic deletion method, for its extremal number $textrm {ex}(n,H)$, the maximum number of edges in an $n$-vertex $H$-free $r$-uniform hypergraph. With this application in mind, we find a range of new counterexamples to the conjecture for hypergraphs, including all linear hypergraphs containing a loose triangle and all $3$-partite $3$-uniform tight cycles.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps ℙk（ℂ）和多项式相似映射的内定态全态族中的强概率稳定性
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-24 DOI: 10.1093/imrn/rnae081
Fabrizio Bianchi, K. Rakhimov
{"title":"Strong Probabilistic Stability in Holomorphic Families of Endomorphisms of ℙk (ℂ) and Polynomial-Like Maps","authors":"Fabrizio Bianchi, K. Rakhimov","doi":"10.1093/imrn/rnae081","DOIUrl":"https://doi.org/10.1093/imrn/rnae081","url":null,"abstract":"\u0000 We prove that, in stable families of endomorphisms of $mathbb{P}^{k} (mathbb C)$, the measurable holomorphic motion of the Julia sets introduced by Berteloot, Dupont, and the first author is unbranched at almost every point with respect to all measures on the Julia set with strictly positive Lyapunov exponents and not charging the post-critical set. This provides a parallel in this setting to the probabilistic stability of Hénon maps by Berger–Dujardin–Lyubich. An analogous result holds in families of polynomial-like maps of large topological degree. In this case, we also give a sufficient condition for the positivity of the Lyapunov exponents of an ergodic measure in terms of its measure-theoretic entropy, generalizing to this setting an analogous result by de Thélin and Dupont valid on $mathbb{P}^{k} (mathbb C)$.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140659164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Log-Concavity of the Alexander Polynomial 亚历山大多项式的对数凹性
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-23 DOI: 10.1093/imrn/rnae058
Elena S Hafner, Karola Mészáros, Alexander Vidinas
{"title":"Log-Concavity of the Alexander Polynomial","authors":"Elena S Hafner, Karola Mészáros, Alexander Vidinas","doi":"10.1093/imrn/rnae058","DOIUrl":"https://doi.org/10.1093/imrn/rnae058","url":null,"abstract":"The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial $Delta _{L}(t)$ of an alternating link $L$ are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus $2$ alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of $Delta _{L}(t)$, where $L$ is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804359","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multimatroids and Rational Curves with Cyclic Action 具有循环作用的多面体和有理曲线
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-23 DOI: 10.1093/imrn/rnae069
Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Shiyue Li
{"title":"Multimatroids and Rational Curves with Cyclic Action","authors":"Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Shiyue Li","doi":"10.1093/imrn/rnae069","DOIUrl":"https://doi.org/10.1093/imrn/rnae069","url":null,"abstract":"We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806493","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
A Construction of Deformations to General Algebras 通用代数的变形构造
IF 1 2区 数学
International Mathematics Research Notices Pub Date : 2024-04-22 DOI: 10.1093/imrn/rnae077
David Bowman, Dora Puljić, Agata Smoktunowicz
{"title":"A Construction of Deformations to General Algebras","authors":"David Bowman, Dora Puljić, Agata Smoktunowicz","doi":"10.1093/imrn/rnae077","DOIUrl":"https://doi.org/10.1093/imrn/rnae077","url":null,"abstract":"One of the questions investigated in deformation theory is to determine to which algebras can a given associative algebra be deformed. In this paper we investigate a different but related question, namely: for a given associative finite-dimensional ${mathbb{C}}$-algebra $A$, find algebras $N$, which can be deformed to $A$. We develop a simple method that produces associative and flat deformations to investigate this question. As an application of this method we answer a question of Michael Wemyss about deformations of contraction algebras.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140804358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","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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