Pub Date : 2022-06-29DOI: 10.1007/s40598-022-00205-2
Toshizumi Fukui, Takeki Tsuchiya
We discuss the notion of properness of a polynomial map (varvec{f}:mathbb {K}^mrightarrow mathbb {K}^n), (mathbb {K}=mathbb {C}) or (mathbb {R}), at a point of the target. We present a method to describe the set of non-proper points of (varvec{f}) with respect to Newton polyhedra of (varvec{f}). We obtain an explicit precise description of such a set of (varvec{f}) when (varvec{f}) 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 (varvec{f}) without the condition (1.5) is also given in Sect. 5.
{"title":"Properness of Polynomial Maps with Newton Polyhedra","authors":"Toshizumi Fukui, 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}
Pub Date : 2022-06-27DOI: 10.1007/s40598-022-00208-z
Avery St. Dizier, Alexander Yong
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.
{"title":"Generalized Permutahedra and Schubert Calculus","authors":"Avery St. Dizier, 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}
Pub Date : 2022-05-27DOI: 10.1007/s40598-022-00200-7
Trevor Clark, Kostiantyn Drach, Oleg Kozlovski, Sebastian van Strien
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: