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+...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
| Published: |
2005
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| Summary: | 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. |
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