{"title":"对Yomdin–Gromov代数引理的再认识","authors":"Gal Binyamini, Dmitry Novikov","doi":"10.1007/s40598-021-00176-w","DOIUrl":null,"url":null,"abstract":"<div><p>In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for <span>\\(C^\\infty \\)</span> maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be <i>cellular</i>. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.</p></div>","PeriodicalId":37546,"journal":{"name":"Arnold Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2021-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40598-021-00176-w","citationCount":"4","resultStr":"{\"title\":\"The Yomdin–Gromov Algebraic Lemma Revisited\",\"authors\":\"Gal Binyamini, Dmitry Novikov\",\"doi\":\"10.1007/s40598-021-00176-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for <span>\\\\(C^\\\\infty \\\\)</span> maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be <i>cellular</i>. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.</p></div>\",\"PeriodicalId\":37546,\"journal\":{\"name\":\"Arnold Mathematical Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-05-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1007/s40598-021-00176-w\",\"citationCount\":\"4\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Arnold Mathematical Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s40598-021-00176-w\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Mathematics\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Arnold Mathematical Journal","FirstCategoryId":"1085","ListUrlMain":"https://link.springer.com/article/10.1007/s40598-021-00176-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
In 1987, Yomdin proved a lemma on smooth parametrizations of semialgebraic sets as part of his solution of Shub’s entropy conjecture for \(C^\infty \) maps. The statement was further refined by Gromov, producing what is now known as the Yomdin–Gromov algebraic lemma. Several complete proofs based on Gromov’s sketch have appeared in the literature, but these have been considerably more complicated than Gromov’s original presentation due to some technical issues. In this note, we give a proof that closely follows Gromov’s original presentation. We prove a somewhat stronger statement, where the parametrizing maps are guaranteed to be cellular. It turns out that this additional restriction, along with some elementary lemmas on differentiable functions in o-minimal structures, allows the induction to be carried out without technical difficulties.
期刊介绍：
The Arnold Mathematical Journal publishes interesting and understandable results in all areas of mathematics. The name of the journal is not only a dedication to the memory of Vladimir Arnold (1937 – 2010), one of the most influential mathematicians of the 20th century, but also a declaration that the journal should serve to maintain and promote the scientific style characteristic for Arnold''s best mathematical works. Features of AMJ publications include: Popularity. The journal articles should be accessible to a very wide community of mathematicians. Not only formal definitions necessary for the understanding must be provided but also informal motivations even if the latter are well-known to the experts in the field. Interdisciplinary and multidisciplinary mathematics. AMJ publishes research expositions that connect different mathematical subjects. Connections that are useful in both ways are of particular importance. Multidisciplinary research (even if the disciplines all belong to pure mathematics) is generally hard to evaluate, for this reason, this kind of research is often under-represented in specialized mathematical journals. AMJ will try to compensate for this.Problems, objectives, work in progress. Most scholarly publications present results of a research project in their “final'' form, in which all posed questions are answered. Some open questions and conjectures may be even mentioned, but the very process of mathematical discovery remains hidden. Following Arnold, publications in AMJ will try to unhide this process and made it public by encouraging the authors to include informal discussion of their motivation, possibly unsuccessful lines of attack, experimental data and close by research directions. AMJ publishes well-motivated research problems on a regular basis. Problems do not need to be original; an old problem with a new and exciting motivation is worth re-stating. Following Arnold''s principle, a general formulation is less desirable than the simplest partial case that is still unknown.Being interesting. The most important requirement is that the article be interesting. It does not have to be limited by original research contributions of the author; however, the author''s responsibility is to carefully acknowledge the authorship of all results. Neither does the article need to consist entirely of formal and rigorous arguments. It can contain parts, in which an informal author''s understanding of the overall picture is presented; however, these parts must be clearly indicated.
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