Zeros of systems of p-adic quadratic forms
We show that a system of r quadratic forms over a -adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer.J.Math.87 (1...
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| Formato: | Journal article |
| Idioma: | English |
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2010
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| _version_ | 1826260218760134656 |
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| author | Heath-Brown, D |
| author_facet | Heath-Brown, D |
| author_sort | Heath-Brown, D |
| collection | OXFORD |
| description | We show that a system of r quadratic forms over a -adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer.J.Math.87 (1965), 605-630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a p-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry. © 2010 Foundation Compositio Mathematica. |
| first_indexed | 2024-03-06T19:02:10Z |
| format | Journal article |
| id | oxford-uuid:13e7a321-b93a-49e1-8e3e-3079f296cf5e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T19:02:10Z |
| publishDate | 2010 |
| record_format | dspace |
| spelling | oxford-uuid:13e7a321-b93a-49e1-8e3e-3079f296cf5e2022-03-26T10:16:33ZZeros of systems of p-adic quadratic formsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:13e7a321-b93a-49e1-8e3e-3079f296cf5eEnglishSymplectic Elements at Oxford2010Heath-Brown, DWe show that a system of r quadratic forms over a -adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer.J.Math.87 (1965), 605-630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a p-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry. © 2010 Foundation Compositio Mathematica. |
| spellingShingle | Heath-Brown, D Zeros of systems of p-adic quadratic forms |
| title | Zeros of systems of p-adic quadratic forms |
| title_full | Zeros of systems of p-adic quadratic forms |
| title_fullStr | Zeros of systems of p-adic quadratic forms |
| title_full_unstemmed | Zeros of systems of p-adic quadratic forms |
| title_short | Zeros of systems of p-adic quadratic forms |
| title_sort | zeros of systems of p adic quadratic forms |
| work_keys_str_mv | AT heathbrownd zerosofsystemsofpadicquadraticforms |