{"title":"准线性二次型的扩展卡尔彭科定理和卡尔彭科-梅尔库热夫定理","authors":"Stephen Scully","doi":"arxiv-2409.02059","DOIUrl":null,"url":null,"abstract":"Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$\nof characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar\nextension to the function field of the affine quadric with equation $p=0$. In\nthis article, we establish a strong constraint on $i$ in terms of the dimension\nof $q$ and two stable birational invariants of $p$, one of which is the\nwell-known \"Izhboldin dimension\", and the other of which is a new invariant\nthat we denote $\\Delta(p)$. Examining the contribution from the Izhboldin\ndimension, we obtain a result that unifies and extends the quasilinear\nanalogues of two fundamental results on the isotropy of non-singular quadratic\nforms over function fields of quadrics in arbitrary characteristic due to\nKarpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the\nquasilinear case of a general conjecture previously formulated by the author,\nsuggesting that a substantial refinement of this conjecture should hold.","PeriodicalId":501136,"journal":{"name":"arXiv - MATH - Rings and Algebras","volume":"24 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms\",\"authors\":\"Stephen Scully\",\"doi\":\"arxiv-2409.02059\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$\\nof characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar\\nextension to the function field of the affine quadric with equation $p=0$. In\\nthis article, we establish a strong constraint on $i$ in terms of the dimension\\nof $q$ and two stable birational invariants of $p$, one of which is the\\nwell-known \\\"Izhboldin dimension\\\", and the other of which is a new invariant\\nthat we denote $\\\\Delta(p)$. Examining the contribution from the Izhboldin\\ndimension, we obtain a result that unifies and extends the quasilinear\\nanalogues of two fundamental results on the isotropy of non-singular quadratic\\nforms over function fields of quadrics in arbitrary characteristic due to\\nKarpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the\\nquasilinear case of a general conjecture previously formulated by the author,\\nsuggesting that a substantial refinement of this conjecture should hold.\",\"PeriodicalId\":501136,\"journal\":{\"name\":\"arXiv - MATH - Rings and Algebras\",\"volume\":\"24 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\":\"arXiv - MATH - Rings and Algebras\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.02059\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Rings and Algebras","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.02059","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms
Let $p$ and $q$ be anisotropic quasilinear quadratic forms over a field $F$
of characteristic $2$, and let $i$ be the isotropy index of $q$ after scalar
extension to the function field of the affine quadric with equation $p=0$. In
this article, we establish a strong constraint on $i$ in terms of the dimension
of $q$ and two stable birational invariants of $p$, one of which is the
well-known "Izhboldin dimension", and the other of which is a new invariant
that we denote $\Delta(p)$. Examining the contribution from the Izhboldin
dimension, we obtain a result that unifies and extends the quasilinear
analogues of two fundamental results on the isotropy of non-singular quadratic
forms over function fields of quadrics in arbitrary characteristic due to
Karpenko and Karpenko-Merkurjev, respectively. This proves in a strong way the
quasilinear case of a general conjecture previously formulated by the author,
suggesting that a substantial refinement of this conjecture should hold.