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.
给定 $n in mathbb{N}$，如果存在 $Pin mathbb{C}[y_{1}、dots , y_{n-1}]$ 中存在 $P 和小数指数的单项式 $m_{1}, dots , m_{n-1}$，使得 $F = P(m_{1}, dots , m_{n-1})$ 退化。我们的主要结果表明，每当阶数为 $d geq 1$ 的多项式 $F$ 是非退化的，那么对于每一个有限非空集 $Asubset mathbb{C}$ ，使得 $|Acdot A| leq K|A|$，都有 $$ begin{align*} &；|F(A, dots, A)| gg |A|^{n} 2^{-O_{d,n}((log 2K)^{3 + o(1)})}.end{align*}$$This is sharp since for every degenerate $F$ and finite set $A subset mathbb{C}$ with $|Acdot A| leq K|A|$, one has $$ begin{align*} & |F(A,dots,A)| ll K^{O_{F}(1)}|A|^{n-1}.end{align*}.$$我们的技术依赖于 Freiman 型逆定理和施密特子空间定理。
{"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}
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.
{"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}
Indranil Biswas, Swarnava Mukhopadhyay, Richard Wentworth
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.
{"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}
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.
{"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}
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 $f(K)$.
{"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}
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.
{"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}
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 $mathbb{P}^{k} (mathbb C)$.
{"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}
Elena S Hafner, Karola Mészáros, Alexander Vidinas
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.
{"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}
Emily Clader, Chiara Damiolini, Christopher Eur, Daoji Huang, Shiyue Li
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.
{"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}
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.
{"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}
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