{"title":"实平面多项式映射和叶形的注入性","authors":"","doi":"10.1016/j.na.2024.113645","DOIUrl":null,"url":null,"abstract":"<div><p>We develop tools to count the connected components of the fibers of a polynomial submersion in two real variables <span><math><mi>p</mi></math></span>. As a consequence, we get a necessary condition for a real number to be a bifurcation value of <span><math><mi>p</mi></math></span>. We further present new methods to verify that <span><math><mi>p</mi></math></span> has no Jacobian mates. These results are applied to prove that a polynomial local self-diffeomorphism of the real plane having one coordinate function with degree less than 6 is globally injective. As a byproduct, we completely classify the foliations defined by polynomial submersions of degree less than 6.</p></div>","PeriodicalId":49749,"journal":{"name":"Nonlinear Analysis-Theory Methods & Applications","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0362546X24001640/pdfft?md5=b3a4aff57d2e2c1abcfc70f1614479b1&pid=1-s2.0-S0362546X24001640-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Injectivity of polynomial maps and foliations in the real plane\",\"authors\":\"\",\"doi\":\"10.1016/j.na.2024.113645\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We develop tools to count the connected components of the fibers of a polynomial submersion in two real variables <span><math><mi>p</mi></math></span>. As a consequence, we get a necessary condition for a real number to be a bifurcation value of <span><math><mi>p</mi></math></span>. We further present new methods to verify that <span><math><mi>p</mi></math></span> has no Jacobian mates. These results are applied to prove that a polynomial local self-diffeomorphism of the real plane having one coordinate function with degree less than 6 is globally injective. As a byproduct, we completely classify the foliations defined by polynomial submersions of degree less than 6.</p></div>\",\"PeriodicalId\":49749,\"journal\":{\"name\":\"Nonlinear Analysis-Theory Methods & Applications\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2024-08-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001640/pdfft?md5=b3a4aff57d2e2c1abcfc70f1614479b1&pid=1-s2.0-S0362546X24001640-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Nonlinear Analysis-Theory Methods & Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0362546X24001640\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Nonlinear Analysis-Theory Methods & Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0362546X24001640","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
因此,我们得到了一个实数是 p 的分叉值的必要条件。我们进一步提出了验证 p 没有雅各布队列的新方法。我们应用这些结果证明了实平面上一个坐标函数的多项式局部自变形的阶数小于 6 是全局注入的。作为副产品,我们对由阶数小于 6 的多项式淹没所定义的叶形进行了完全分类。
Injectivity of polynomial maps and foliations in the real plane
We develop tools to count the connected components of the fibers of a polynomial submersion in two real variables . As a consequence, we get a necessary condition for a real number to be a bifurcation value of . We further present new methods to verify that has no Jacobian mates. These results are applied to prove that a polynomial local self-diffeomorphism of the real plane having one coordinate function with degree less than 6 is globally injective. As a byproduct, we completely classify the foliations defined by polynomial submersions of degree less than 6.
期刊介绍:
Nonlinear Analysis focuses on papers that address significant problems in Nonlinear Analysis that have a sustainable and important impact on the development of new directions in the theory as well as potential applications. Review articles on important topics in Nonlinear Analysis are welcome as well. In particular, only papers within the areas of specialization of the Editorial Board Members will be considered. Authors are encouraged to check the areas of expertise of the Editorial Board in order to decide whether or not their papers are appropriate for this journal. The journal aims to apply very high standards in accepting papers for publication.