{"title":"复多项式的莫尔斯数","authors":"Laurenţiu Maxim, Mihai Tibăr","doi":"10.1112/topo.12362","DOIUrl":null,"url":null,"abstract":"<p>To a complex polynomial function <span></span><math>\n <semantics>\n <mi>f</mi>\n <annotation>$f$</annotation>\n </semantics></math> with arbitrary singularities, we associate the number of Morse points in a general linear Morsification <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>f</mi>\n <mi>t</mi>\n </msub>\n <mo>:</mo>\n <mo>=</mo>\n <mi>f</mi>\n <mo>−</mo>\n <mi>t</mi>\n <mi>ℓ</mi>\n </mrow>\n <annotation>$f_{t}:= f - t\\ell$</annotation>\n </semantics></math>. We produce computable algebraic formulae in terms of invariants of <span></span><math>\n <semantics>\n <mi>f</mi>\n <annotation>$f$</annotation>\n </semantics></math> for the numbers of stratwise Morse trajectories that abut, as <span></span><math>\n <semantics>\n <mrow>\n <mi>t</mi>\n <mo>→</mo>\n <mn>0</mn>\n </mrow>\n <annotation>$t\\rightarrow 0$</annotation>\n </semantics></math>, to some point of the space or at infinity.</p>","PeriodicalId":56114,"journal":{"name":"Journal of Topology","volume":"17 4","pages":""},"PeriodicalIF":0.8000,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12362","citationCount":"0","resultStr":"{\"title\":\"Morse numbers of complex polynomials\",\"authors\":\"Laurenţiu Maxim, Mihai Tibăr\",\"doi\":\"10.1112/topo.12362\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>To a complex polynomial function <span></span><math>\\n <semantics>\\n <mi>f</mi>\\n <annotation>$f$</annotation>\\n </semantics></math> with arbitrary singularities, we associate the number of Morse points in a general linear Morsification <span></span><math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>f</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>:</mo>\\n <mo>=</mo>\\n <mi>f</mi>\\n <mo>−</mo>\\n <mi>t</mi>\\n <mi>ℓ</mi>\\n </mrow>\\n <annotation>$f_{t}:= f - t\\\\ell$</annotation>\\n </semantics></math>. We produce computable algebraic formulae in terms of invariants of <span></span><math>\\n <semantics>\\n <mi>f</mi>\\n <annotation>$f$</annotation>\\n </semantics></math> for the numbers of stratwise Morse trajectories that abut, as <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>t</mi>\\n <mo>→</mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$t\\\\rightarrow 0$</annotation>\\n </semantics></math>, to some point of the space or at infinity.</p>\",\"PeriodicalId\":56114,\"journal\":{\"name\":\"Journal of Topology\",\"volume\":\"17 4\",\"pages\":\"\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2024-10-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/topo.12362\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Topology\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/topo.12362\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Topology","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/topo.12362","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
对于具有任意奇异点的复多项式函数 f $f$，我们将一般线性莫尔斯化 f t : = f - t ℓ $f_{t}:= f - t\ell$ 中的莫尔斯点数联系起来。当 t → 0 $t\rightarrow 0$ 时，我们用 f $f$ 的不变量来计算与空间的某个点或无穷远处相交的平分莫尔斯轨迹的数目，从而得出可计算的代数式。
To a complex polynomial function with arbitrary singularities, we associate the number of Morse points in a general linear Morsification . We produce computable algebraic formulae in terms of invariants of for the numbers of stratwise Morse trajectories that abut, as , to some point of the space or at infinity.
期刊介绍：
The Journal of Topology publishes papers of high quality and significance in topology, geometry and adjacent areas of mathematics. Interesting, important and often unexpected links connect topology and geometry with many other parts of mathematics, and the editors welcome submissions on exciting new advances concerning such links, as well as those in the core subject areas of the journal.
The Journal of Topology was founded in 2008. It is published quarterly with articles published individually online prior to appearing in a printed issue.
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