Counting rational points on hypersurfaces
For any n ≧ 2, let F ε ℤ[x1, . . . , xn] be a form of degree d ≧ 2, which produces a geometrically irreducible hypersurface in ℙn-1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F;B) = O(Bn-2+...
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| Format: | Journal article |
| Language: | English |
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2005
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| _version_ | 1797065038896300032 |
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| author | Browning, T Heath-Brown, D |
| author_facet | Browning, T Heath-Brown, D |
| author_sort | Browning, T |
| collection | OXFORD |
| description | For any n ≧ 2, let F ε ℤ[x1, . . . , xn] be a form of degree d ≧ 2, which produces a geometrically irreducible hypersurface in ℙn-1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F;B) = O(Bn-2+ε), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε. © Walter de Gruyter 2005. |
| first_indexed | 2024-03-06T21:22:55Z |
| format | Journal article |
| id | oxford-uuid:42199398-94c0-464d-8898-3903623cae11 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T21:22:55Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:42199398-94c0-464d-8898-3903623cae112022-03-26T14:47:27ZCounting rational points on hypersurfacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:42199398-94c0-464d-8898-3903623cae11EnglishSymplectic Elements at Oxford2005Browning, THeath-Brown, DFor any n ≧ 2, let F ε ℤ[x1, . . . , xn] be a form of degree d ≧ 2, which produces a geometrically irreducible hypersurface in ℙn-1. This paper is concerned with the number N(F;B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F;B) = O(Bn-2+ε), whenever either n ≦ 5 or the hypersurface is not a union of lines. Here the implied constant depends at most upon d, n and ε. © Walter de Gruyter 2005. |
| spellingShingle | Browning, T Heath-Brown, D Counting rational points on hypersurfaces |
| title | Counting rational points on hypersurfaces |
| title_full | Counting rational points on hypersurfaces |
| title_fullStr | Counting rational points on hypersurfaces |
| title_full_unstemmed | Counting rational points on hypersurfaces |
| title_short | Counting rational points on hypersurfaces |
| title_sort | counting rational points on hypersurfaces |
| work_keys_str_mv | AT browningt countingrationalpointsonhypersurfaces AT heathbrownd countingrationalpointsonhypersurfaces |