The Density of Rational Points on Non‐Singular Hypersurfaces, I
The Density of Rational Points on Non‐Singular Hypersurfaces, I
复制标题
非奇异超曲面上有理点的密度，I
DOI：
10.1112/s0024609305018412
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发表时间：
2005
影响因子：
0.9
通讯作者：
D. R. Heath
中科院分区：
文献类型：
--
作者：
Tim D Browning;D. R. Heath
对于任意n <$3，设F ∈ Z[X 0，...，Xn]是定义非奇异超曲面X <$Pn的次数d <$5的形式。本文的主要结果是证明了X上高度至多为B的Q有理点的个数N（F; B）满足N（F; B）=Od，n（Bn−1+ n），对任意n> 0.这个估计中隐含的常数最多取决于d、n和n。新的估计也得到了一个正整数的代表作为三个d次幂的总和的数量，并为缺乏整数解决方案，以平等的总和一样的多项式。2000年数学学科分类11 G35（小学）、11 P05、14 G 05（中学）。
For any n ⩾ 3, let F ∈ Z[X0, …, Xn] be a form of degree d ⩾ 5 that defines a non‐singular hypersurface X ⊂ Pn. The main result in this paper is a proof of the fact that the number N(F; B) of Q‐rational points on X which have height at most B satisfies N(F; B)=Od,ɛ,n(Bn−1+ɛ) , for any ɛ > 0. The implied constant in this estimate depends at most upon d, ɛ and n. New estimates are also obtained for the number of representations of a positive integer as the sum of three dth powers, and for the paucity of integer solutions to equal sums of like polynomials. 2000 Mathematics Subject Classification 11G35 (primary), 11P05, 14G05 (secondary).