Arnold Mathematical Journal最新文献

{"title":"Properness of Polynomial Maps with Newton Polyhedra","authors":"Toshizumi Fukui,&nbsp;Takeki Tsuchiya","doi":"10.1007/s40598-022-00205-2","DOIUrl":"10.1007/s40598-022-00205-2","url":null,"abstract":"<div><p>We discuss the notion of properness of a polynomial map <span>(varvec{f}:mathbb {K}^mrightarrow mathbb {K}^n)</span>, <span>(mathbb {K}=mathbb {C})</span> or <span>(mathbb {R})</span>, at a point of the target. We present a method to describe the set of non-proper points of <span>(varvec{f})</span> with respect to Newton polyhedra of <span>(varvec{f})</span>. We obtain an explicit precise description of such a set of <span>(varvec{f})</span> when <span>(varvec{f})</span> satisfies certain condition (1.5). A relative version is also given in Sect. 3. Several tricks to describe the set of non-proper points of <span>(varvec{f})</span> without the condition (1.5) is also given in Sect. 5.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43371287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generalized Permutahedra and Schubert Calculus","authors":"Avery St. Dizier,&nbsp;Alexander Yong","doi":"10.1007/s40598-022-00208-z","DOIUrl":"10.1007/s40598-022-00208-z","url":null,"abstract":"<div><p>We connect generalized permutahedra with Schubert calculus. Thereby, we give sufficient vanishing criteria for Schubert intersection numbers of the flag variety. Our argument utilizes recent developments in the study of Schubitopes, which are Newton polytopes of Schubert polynomials. The resulting tableau test executes in polynomial time.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46821283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Dynamics of Complex Box Mappings","authors":"Trevor Clark,&nbsp;Kostiantyn Drach,&nbsp;Oleg Kozlovski,&nbsp;Sebastian van Strien","doi":"10.1007/s40598-022-00200-7","DOIUrl":"10.1007/s40598-022-00200-7","url":null,"abstract":"<div><p>In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is:</p><ul>\u0000 <li>\u0000 <p>To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues.</p>\u0000 </li>\u0000 <li>\u0000 <p>To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings <i>dynamically natural</i>. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics.</p>\u0000 </li>\u0000 <li>\u0000 <p>Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are:</p><ul>\u0000 <li>\u0000 <p>the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (Ann Math 165:749–841, 2007), referred to below as KSS;</p>\u0000 </li>\u0000 <li>\u0000 <p>the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009);</p>\u0000 </li>\u0000 <li>\u0000 <p>the QC-Criterion and the Spreading Principle from KSS.</p>\u0000 </li>\u0000 </ul><p> The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings.</p>\u0000 </li>\u0000 <li>\u0000 <p>To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings:</p><ul>\u0000 <li>\u0000 <p>puzzle pieces shrink to points,</p>\u0000 </li>\u0000 <li>\u0000 <p>(under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate.</p>\u0000 </li>\u0000 </ul>\u0000 </li>\u0000 <li>\u0000 <p>We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting.</p>\u0000 </li>\u0000 <li>\u0000 ","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00200-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50518555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Maximum Likelihood Degree of Surjective Rational Maps","authors":"Ilya Karzhemanov","doi":"10.1007/s40598-022-00207-0","DOIUrl":"10.1007/s40598-022-00207-0","url":null,"abstract":"<div><p>With any <i>surjective rational map</i> <span>(f: mathbb {P}^n dashrightarrow mathbb {P}^n)</span> of the projective space, we associate a numerical invariant (<i>ML degree</i>) and compute it in terms of a naturally defined vector bundle <span>(E_f longrightarrow mathbb {P}^n)</span>.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45581553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Classification of Generic Spherical Quadrilaterals","authors":"Andrei Gabrielov","doi":"10.1007/s40598-022-00204-3","DOIUrl":"10.1007/s40598-022-00204-3","url":null,"abstract":"<div><p>Generic spherical quadrilaterals are classified up to isometry. Condition of genericity consists in the requirement that the images of the sides under the developing map belong to four distinct circles which have no triple intersections. Under this condition, it is shown that the space of quadrilaterals with prescribed angles consists of finitely many open curves. Degeneration at the endpoints of these curves is also determined.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40598-022-00204-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42828660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cohomology Rings and Algebraic Torus Actions on Hypersurfaces in the Product of Projective Spaces and Bounded Flag Varieties","authors":"Grigory Solomadin","doi":"10.1007/s40598-022-00203-4","DOIUrl":"10.1007/s40598-022-00203-4","url":null,"abstract":"<div><p>In this paper, for any Milnor hypersurface, we find the largest dimension of effective algebraic torus actions on it. The proof of the corresponding theorem is based on the computation of the automorphism group for any Milnor hypersurface. We find all generalized Buchstaber–Ray and Ray hypersurfaces that are toric varieties. We compute the Betti numbers of these hypersurfaces and describe their integral singular cohomology rings in terms of the cohomology of the corresponding ambient varieties.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46354160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Quantitative Uncertainty Principles Related to Lions Transform","authors":"A. Achak,&nbsp;A. Abouelaz,&nbsp;R. Daher,&nbsp;N. Safouane","doi":"10.1007/s40598-022-00202-5","DOIUrl":"10.1007/s40598-022-00202-5","url":null,"abstract":"<div><p>We prove various mathematical aspects of the quantitative uncertainty principles, including Donoho–Stark’s uncertainty principle and a variant of Benedicks theorem for Lions transform.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43801122","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-Sided Fundamental Theorem of Affine Geometry","authors":"Alexey Gorinov","doi":"10.1007/s40598-022-00201-6","DOIUrl":"10.1007/s40598-022-00201-6","url":null,"abstract":"<div><p>The fundamental theorem of affine geometry says that if a self-bijection <i>f</i> of an affine space of dimenion <i>n</i> over a possibly skew field takes left affine subspaces to left affine subspaces of the same dimension, then <i>f</i> of the expected type, namely <i>f</i> is a composition of an affine map and an automorphism of the field. We prove a two-sided analogue of this: namely, we consider self-bijections as above which take affine subspaces to affine subspaces but which are allowed to take left subspaces to right ones and vice versa. We show that under some conditions these maps again are of the expected type.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41287513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Renormalization of Bicritical Circle Maps","authors":"Gabriela Estevez,&nbsp;Pablo Guarino","doi":"10.1007/s40598-022-00199-x","DOIUrl":"10.1007/s40598-022-00199-x","url":null,"abstract":"<div><p>A general <i>ansatz</i> in Renormalization Theory, already established in many important situations, states that exponential convergence of renormalization orbits implies that topological conjugacies are actually smooth (when restricted to the attractors of the original systems). In this paper, we establish this principle for a large class of <i>bicritical circle maps</i>, which are <span>(C^3)</span> circle homeomorphisms with irrational rotation number and exactly two (non-flat) critical points. The proof presented here is an adaptation, to the bicritical setting, of the one given by de Faria and de Melo in (J Eur Math Soc 1:339–392, 1999) for the case of a single critical point. When combined with the recent papers (Estevez et al. in Complex bounds for multicritical circle maps with bounded type rotation number, arXiv:2005.02377, 2020; Yampolsky in C R Math Rep Acad Sci Can 41:57–83, 2019), our main theorem implies <span>(C^{1+alpha })</span> rigidity for real-analytic bicritical circle maps with rotation number of <i>bounded type</i> (Corollary 1.1).</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49563940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical Moduli Spaces and Polynomial Endomorphisms of Configurations","authors":"Talia Blum,&nbsp;John R. Doyle,&nbsp;Trevor Hyde,&nbsp;Colby Kelln,&nbsp;Henry Talbott,&nbsp;Max Weinreich","doi":"10.1007/s40598-022-00197-z","DOIUrl":"10.1007/s40598-022-00197-z","url":null,"abstract":"<div><p>A portrait is a combinatorial model for a discrete dynamical system on a finite set. We study the geometry of portrait moduli spaces, whose points correspond to equivalence classes of point configurations on the affine line for which there exist polynomials realizing the dynamics of a given portrait. We present results and pose questions inspired by a large-scale computational survey of intersections of portrait moduli spaces for polynomials in low degree.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47885513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}