{"title":"阿尔廷-施莱尔扩展与亨塞尔有价域的组合复杂性","authors":"BLAISE BOISSONNEAU","doi":"10.1017/jsl.2024.34","DOIUrl":null,"url":null,"abstract":"<p>We give explicit formulas witnessing IP, IP<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$_{\\!n}$</span></span></img></span></span>, or TP2 in fields with Artin–Schreier extensions. We use them to control <span>p</span>-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$_{\\!n}$</span></span></img></span></span> context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$_{\\!n}$</span></span></img></span></span> henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.</p>","PeriodicalId":501300,"journal":{"name":"The Journal of Symbolic Logic","volume":"16 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"ARTIN–SCHREIER EXTENSIONS AND COMBINATORIAL COMPLEXITY IN HENSELIAN VALUED FIELDS\",\"authors\":\"BLAISE BOISSONNEAU\",\"doi\":\"10.1017/jsl.2024.34\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We give explicit formulas witnessing IP, IP<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{\\\\!n}$</span></span></img></span></span>, or TP2 in fields with Artin–Schreier extensions. We use them to control <span>p</span>-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{\\\\!n}$</span></span></img></span></span> context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240911114232841-0471:S0022481224000343:S0022481224000343_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{\\\\!n}$</span></span></img></span></span> henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.</p>\",\"PeriodicalId\":501300,\"journal\":{\"name\":\"The Journal of Symbolic Logic\",\"volume\":\"16 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Journal of Symbolic Logic\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1017/jsl.2024.34\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Journal of Symbolic Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1017/jsl.2024.34","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
ARTIN–SCHREIER EXTENSIONS AND COMBINATORIAL COMPLEXITY IN HENSELIAN VALUED FIELDS
We give explicit formulas witnessing IP, IP$_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP$_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP$_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.
$_{\!n}$, or TP2 in fields with Artin–Schreier extensions. We use them to control p-extensions of mixed characteristic henselian valued fields, allowing us most notably to generalize to the NIP
$_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP
$_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.
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