Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$
For $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the...
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
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| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.668/ |
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| author | Harvey, Curtis R. Karpenko, Nikita A. |
| author_facet | Harvey, Curtis R. Karpenko, Nikita A. |
| author_sort | Harvey, Curtis R. |
| collection | DOAJ |
| description | For $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the corresponding quadratic forms up to odd degree base field extensions and get this way a strong upper bound on their essential $2$-dimension. As well, we compute the reduced Chow group of the maximal orthogonal grassmannian of the quadratic form and conclude that its canonical $2$-dimension is $2^n+2^{n-2}-2$. |
| format | Article |
| id | doaj-art-f64c4c517951414d8ca113f5aee4d870 |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-f64c4c517951414d8ca113f5aee4d8702025-02-07T11:23:51ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G111539154310.5802/crmath.66810.5802/crmath.668Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$Harvey, Curtis R.0Karpenko, Nikita A.1Mathematical & Statistical Sciences, University of Alberta, Edmonton, CanadaMathematical & Statistical Sciences, University of Alberta, Edmonton, CanadaFor $n\ge 3$, confirming a weak version of a conjecture of Hoffmann, we show that every anisotropic quadratic form in $I^n$ of dimension $2^n+2^{n-1}$ splits over a finite extension of the base field of degree not divisible by $4$. The first new case is $n=4$, where we obtain a classification of the corresponding quadratic forms up to odd degree base field extensions and get this way a strong upper bound on their essential $2$-dimension. As well, we compute the reduced Chow group of the maximal orthogonal grassmannian of the quadratic form and conclude that its canonical $2$-dimension is $2^n+2^{n-2}-2$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.668/Quadratic forms over fieldsprojective homogeneous varietiesChow rings |
| spellingShingle | Harvey, Curtis R. Karpenko, Nikita A. Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ Comptes Rendus. Mathématique Quadratic forms over fields projective homogeneous varieties Chow rings |
| title | Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ |
| title_full | Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ |
| title_fullStr | Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ |
| title_full_unstemmed | Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ |
| title_short | Quadratic forms in $I^n$ of dimension $2^n+2^{n-1}$ |
| title_sort | quadratic forms in i n of dimension 2 n 2 n 1 |
| topic | Quadratic forms over fields projective homogeneous varieties Chow rings |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.668/ |
| work_keys_str_mv | AT harveycurtisr quadraticformsininofdimension2n2n1 AT karpenkonikitaa quadraticformsininofdimension2n2n1 |