Forum of Mathematics Sigma最新文献

{"title":"Lower Bounds for the Canonical Height of a Unicritical Polynomial and Capacity","authors":"P. Habegger, H. Schmidt","doi":"10.1017/fms.2023.112","DOIUrl":"https://doi.org/10.1017/fms.2023.112","url":null,"abstract":"<p>In a recent breakthrough, Dimitrov [Dim] solved the Schinzel–Zassenhaus conjecture. We follow his approach and adapt it to certain dynamical systems arising from polynomials of the form <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$T^p+c$</span></span></img></span></span>, where <span>p</span> is a prime number and where the orbit of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span> is finite. For example, if <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$p=2$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$0$</span></span></img></span></span> is periodic under <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$T^2+c$</span></span></img></span></span> with <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$cin mathbb {R}$</span></span></img></span></span>, we prove a lower bound for the local canonical height of a wandering algebraic integer that is inversely proportional to the field degree. From this, we are able to deduce a lower bound for the canonical height of a wandering point that decays like the inverse square of the field degree. For these <span>f</span>, our method has application to the irreducibility of polynomials. Indeed, say <span>y</span> is preperiodic under <span>f</span> but not periodic. Then any iteration of <span>f</span> minus <span>y</span> is irreducible in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328033452511-0843:S2050509423001123:S2050509423001123_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$mathbb {Q}(y)[T]$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"69 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579080","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}

{"title":"Whittaker categories of quasi-reductive lie superalgebras and quantum symmetric pairs","authors":"Chih-Whi Chen, Shun-Jen Cheng","doi":"10.1017/fms.2024.17","DOIUrl":"https://doi.org/10.1017/fms.2024.17","url":null,"abstract":"<p>We show that, for an arbitrary finite-dimensional quasi-reductive Lie superalgebra over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline1.png\"><span data-mathjax-type=\"texmath\"><span>${mathbb {C}}$</span></span></img></span></span> with a triangular decomposition and a character <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$zeta $</span></span></img></span></span> of the nilpotent radical, the associated Backelin functor <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma _zeta $</span></span></img></span></span> sends Verma modules to standard Whittaker modules provided the latter exist. As a consequence, this gives a complete solution to the problem of determining the composition factors of the standard Whittaker modules in terms of composition factors of Verma modules in the category <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline4.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal {O}}$</span></span></img></span></span>. In the case of the ortho-symplectic Lie superalgebras, we show that the Backelin functor <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$Gamma _zeta $</span></span></img></span></span> and its target category, respectively, categorify a <span>q</span>-symmetrizing map and the corresponding <span>q</span>-symmetrized Fock space associated with a quasi-split quantum symmetric pair of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327170422340-0203:S2050509424000173:S2050509424000173_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$AIII$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"57 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578951","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}

{"title":"Extensions of the colorful Helly theorem for d-collapsible and d-Leray complexes","authors":"Minki Kim, Alan Lew","doi":"10.1017/fms.2024.23","DOIUrl":"https://doi.org/10.1017/fms.2024.23","url":null,"abstract":"&lt;p&gt;We present extensions of the colorful Helly theorem for &lt;span&gt;d&lt;/span&gt;-collapsible and &lt;span&gt;d&lt;/span&gt;-Leray complexes, providing a common generalization to the matroidal versions of the theorem due to Kalai and Meshulam, the ‘very colorful’ Helly theorem introduced by Arocha, Bárány, Bracho, Fabila and Montejano and the ‘semi-intersecting’ colorful Helly theorem proved by Montejano and Karasev.&lt;/p&gt;&lt;p&gt;As an application, we obtain the following extension of Tverberg’s theorem: Let &lt;span&gt;A&lt;/span&gt; be a finite set of points in &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;${mathbb R}^d$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; with &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$|A|&gt;(r-1)(d+1)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. Then, there exist a partition &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$A_1,ldots ,A_r$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of &lt;span&gt;A&lt;/span&gt; and a subset &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$Bsubset A$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; of size &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$(r-1)(d+1)$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; such that &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$cap _{i=1}^r operatorname {mathrm {text {conv}}}( (Bcup {p})cap A_i)neq emptyset $&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; for all &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328020016659-0471:S2050509424000239:S2050509424000239_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$pin Asetminus B$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. That is, we obtain a partition of &lt;span&gt;A&lt;/span&gt; into &lt;span&gt;r&lt;/span&gt; parts that remains a Tverberg partition even after removing all but one arbitrary point from &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridg","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"51 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578950","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}

{"title":"Evaluation of and period polynomial relations","authors":"Steven Charlton, Adam Keilthy","doi":"10.1017/fms.2024.16","DOIUrl":"https://doi.org/10.1017/fms.2024.16","url":null,"abstract":"<p>In studying the depth filtration on multiple zeta values, difficulties quickly arise due to a disparity between it and the coradical filtration [9]. In particular, there are additional relations in the depth graded algebra coming from period polynomials of cusp forms for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$operatorname {mathrm {SL}}_2({mathbb {Z}})$</span></span></img></span></span>. In contrast, a simple combinatorial filtration, the block filtration [13, 28] is known to agree with the coradical filtration, and so there is no similar defect in the associated graded. However, via an explicit evaluation of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$zeta (2,ldots ,2,4,2,ldots ,2)$</span></span></img></span></span> as a polynomial in double zeta values, we derive these period polynomial relations as a consequence of an intrinsic symmetry of block graded multiple zeta values in block degree 2. In deriving this evaluation, we find a Galois descent of certain alternating double zeta values to classical double zeta values, which we then apply to give an evaluation of the multiple <span>t</span> values [22] <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328145638660-0911:S2050509424000161:S2050509424000161_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$t(2ell ,2k)$</span></span></img></span></span> in terms of classical double zeta values.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"4 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140578943","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}

{"title":"Stability of isometric immersions of hypersurfaces","authors":"Itai Alpern, Raz Kupferman, Cy Maor","doi":"10.1017/fms.2024.30","DOIUrl":"https://doi.org/10.1017/fms.2024.30","url":null,"abstract":"<p>We prove a stability result of isometric immersions of hypersurfaces in Riemannian manifolds, with respect to <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$L^p$</span></span></img></span></span>-perturbations of their fundamental forms: For a manifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline2.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal M}^d$</span></span></img></span></span> endowed with a reference metric and a reference shape operator, we show that a sequence of immersions <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$f_n:{mathcal M}^dto {mathcal N}^{d+1}$</span></span></img></span></span>, whose pullback metrics and shape operators are arbitrary close in <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$L^p$</span></span></img></span></span> to the reference ones, converge to an isometric immersion having the reference shape operator. This result is motivated by elasticity theory and generalizes a previous result [AKM22] to a general target manifold <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240328015647453-0017:S2050509424000306:S2050509424000306_inline5.png\"><span data-mathjax-type=\"texmath\"><span>${mathcal N}$</span></span></img></span></span>, removing a constant curvature assumption. The method of proof differs from that in [AKM22]: it extends a Young measure approach that was used in codimension-0 stability results, together with an appropriate relaxation of the energy and a regularity result for immersions satisfying given fundamental forms. In addition, we prove a related quantitative (rather than asymptotic) stability result in the case of Euclidean target, similar to [CMM19] but with no a priori assumed bounds.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579196","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}

{"title":"Cohomological","authors":"Woonam Lim, Miguel Moreira, Weite Pi","doi":"10.1017/fms.2024.31","DOIUrl":"https://doi.org/10.1017/fms.2024.31","url":null,"abstract":"<p>We prove that the cohomology rings of the moduli space <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$M_{d,chi }$</span></span></img></span></span> of one-dimensional sheaves on the projective plane are not isomorphic for general different choices of the Euler characteristics. This stands in contrast to the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$chi $</span></span></img></span></span>-independence of the Betti numbers of these moduli spaces. As a corollary, we deduce that <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240327172850156-0188:S2050509424000318:S2050509424000318_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$M_{d,chi }$</span></span></img></span></span> are topologically different unless they are related by obvious symmetries, strengthening a previous result of Woolf distinguishing them as algebraic varieties.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"17 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140579189","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}

{"title":"K-stable smooth Fano threefolds of Picard rank two","authors":"Ivan Cheltsov, Elena Denisova, Kento Fujita","doi":"10.1017/fms.2024.5","DOIUrl":"https://doi.org/10.1017/fms.2024.5","url":null,"abstract":"<p>We prove that all smooth Fano threefolds in the families <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline1.png?pub-status=live\" type=\"\"> and <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline2.png?pub-status=live\" type=\"\"> are K-stable, and we also prove that smooth Fano threefolds in the family <img mimesubtype=\"png\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240319152445390-0178:S2050509424000057:S2050509424000057_inline3.png?pub-status=live\" type=\"\"> that satisfy one very explicit generality condition are K-stable.</img></img></img></p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"20 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140166669","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}

{"title":"Undecidability of polynomial inequalities in weighted graph homomorphism densities","authors":"Grigoriy Blekherman, Annie Raymond, Fan Wei","doi":"10.1017/fms.2024.19","DOIUrl":"https://doi.org/10.1017/fms.2024.19","url":null,"abstract":"&lt;p&gt;Many problems and conjectures in extremal combinatorics concern polynomial inequalities between homomorphism densities of graphs where we allow edges to have real weights. Using the theory of graph limits, we can equivalently evaluate polynomial expressions in homomorphism densities on &lt;span&gt;kernels W&lt;/span&gt;, that is, symmetric, bounded and measurable functions &lt;span&gt;W&lt;/span&gt; from &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline1.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$[0,1]^2 to mathbb {R}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. In 2011, Hatami and Norin proved a fundamental result that it is undecidable to determine the validity of polynomial inequalities in homomorphism densities for graphons (i.e., the case where the range of &lt;span&gt;W&lt;/span&gt; is &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline2.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$[0,1]$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, which corresponds to unweighted graphs or, equivalently, to graphs with edge weights between &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline3.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$0$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt; and &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline4.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$1$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;). The corresponding problem for more general sets of kernels, for example, for all kernels or for kernels with range &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline5.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$[-1,1]$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, remains open. For any &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline6.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;$a&gt; 0$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;, we show undecidability of polynomial inequalities for any set of kernels which contains all kernels with range &lt;span&gt;&lt;span&gt;&lt;img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240315134900186-0773:S2050509424000197:S2050509424000197_inline7.png\"&gt;&lt;span data-mathjax-type=\"texmath\"&gt;&lt;span&gt;${0,a}$&lt;/span&gt;&lt;/span&gt;&lt;/img&gt;&lt;/span&gt;&lt;/span&gt;. This result also an","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"97 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149592","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}

{"title":"On the Turán number of the hypercube","authors":"Oliver Janzer, Benny Sudakov","doi":"10.1017/fms.2024.27","DOIUrl":"https://doi.org/10.1017/fms.2024.27","url":null,"abstract":"In 1964, Erdős proposed the problem of estimating the Turán number of the &lt;jats:italic&gt;d&lt;/jats:italic&gt;-dimensional hypercube &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline1.png\" /&gt; &lt;jats:tex-math&gt; $Q_d$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. Since &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline2.png\" /&gt; &lt;jats:tex-math&gt; $Q_d$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a bipartite graph with maximum degree &lt;jats:italic&gt;d&lt;/jats:italic&gt;, it follows from results of Füredi and Alon, Krivelevich, Sudakov that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline3.png\" /&gt; &lt;jats:tex-math&gt; $mathrm {ex}(n,Q_d)=O_d(n^{2-1/d})$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. A recent general result of Sudakov and Tomon implies the slightly stronger bound &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline4.png\" /&gt; &lt;jats:tex-math&gt; $mathrm {ex}(n,Q_d)=o(n^{2-1/d})$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. We obtain the first power-improvement for this old problem by showing that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline5.png\" /&gt; &lt;jats:tex-math&gt; $mathrm {ex}(n,Q_d)=O_dleft (n^{2-frac {1}{d-1}+frac {1}{(d-1)2^{d-1}}}right )$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. This answers a question of Liu. Moreover, our techniques give a power improvement for a larger class of graphs than cubes. We use a similar method to prove that any &lt;jats:italic&gt;n&lt;/jats:italic&gt;-vertex, properly edge-coloured graph without a rainbow cycle has at most &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline6.png\" /&gt; &lt;jats:tex-math&gt; $O(n(log n)^2)$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; edges, improving the previous best bound of &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline7.png\" /&gt; &lt;jats:tex-math&gt; $n(log n)^{2+o(1)}$ &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; by Tomon. Furthermore, we show that any properly edge-coloured &lt;jats:italic&gt;n&lt;/jats:italic&gt;-vertex graph with &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S2050509424000276_inline8.png\" /&gt; &lt;jats:tex-math&gt; $omega (nlog n)$ &lt;/jats:te","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"109 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140149744","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}

{"title":"Equivariant Hodge polynomials of heavy/light moduli spaces","authors":"Siddarth Kannan, Stefano Serpente, Claudia He Yun","doi":"10.1017/fms.2024.20","DOIUrl":"https://doi.org/10.1017/fms.2024.20","url":null,"abstract":"<p>Let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$overline {mathcal {M}}_{g, m|n}$</span></span></img></span></span> denote Hassett’s moduli space of weighted pointed stable curves of genus <span>g</span> for the <span>heavy/light</span> weight data <span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_eqnu1.png\"><span data-mathjax-type=\"texmath\"><span>$$begin{align*}left(1^{(m)}, 1/n^{(n)}right),end{align*}$$</span></span></img></span></p><p>and let <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {M}_{g, m|n} subset overline {mathcal {M}}_{g, m|n}$</span></span></img></span></span> be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$(S_mtimes S_n)$</span></span></img></span></span>-equivariant Hodge–Deligne polynomials of these spaces in terms of the generating functions for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$S_{n}$</span></span></img></span></span>-equivariant Hodge–Deligne polynomials of <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$overline {mathcal {M}}_{g,n}$</span></span></img></span></span> and <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240313134052190-0722:S2050509424000203:S2050509424000203_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$mathcal {M}_{g,n}$</span></span></img></span></span>.</p>","PeriodicalId":56000,"journal":{"name":"Forum of Mathematics Sigma","volume":"68 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140125511","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}