Cubic Forms in Ten Variables
Cubic Forms in Ten Variables
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DOI：
10.1112/plms/s3-47.2.225
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发表时间：
1983-09
影响因子：
1.8
通讯作者：
D. R. Heath-Brown
中科院分区：
文献类型：
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作者：
D. R. Heath-Brown
设F（x）= F（x1，...，xn）是具有有理系数的三次形式。证明了F有一个非平凡的有理数零点，只要n^ 10。Mordell [19]已经通过例子证明，如果允许n= 9，则该猜想是错误的。第一个充分条件n^n 0由刘易斯[18]给出。他的方法产生的n 0值在500^n 0 ^ 1000范围内。与刘易斯的证明几乎同时，Birch [1]和Davenport [6]也给出了另一种证明。达文波特的论证产生了n 0 = 32，随后的改进首先将其减少到n 0 = 29 [7]，然后减少到n 0 = 16 [8]。我们将证明最好的可能结果，其中n 0 = i 0，至少当F是非奇异的。这意味着C-{0}上的VF# 0，其中照常。
Let F (x)= F (x1,..., xn) be a cubic form with rational coefficients. It is conjectured that F has a non-trivial rational zero, providing that n^ 10. Mordell [19] has shown by examples that the conjecture would be false if one allowed n= 9. The first sufficient condition n^ n0 was given by Lewis [18]. His method yields a value for n0 in the range 500^ n0^ 1000. Alternative proofs were given almost simultaneously with that of Lewis, by Birch [1] and Davenport [6]. Davenport's argument produces n0= 32, and subsequent improvements reduced this firstly to n0= 29 [7], and then to n0= 16 [8]. We shall prove the best possible result, with n0= i0, at least when F is non-singular. By this we mean that VF# 0 on C—{0}, where as usual.