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Forms in many variables. (English) Zbl 0905.11022

Motohashi, Y. (ed.), Analytic number theory. Proceedings of the 39th Taniguchi international symposium on mathematics, Kyoto, Japan, May 13–17, 1996. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 247, 361-376 (1997).
When \(K\) is a field, \(d\) and \(r\) are natural numbers, and \(m\) is a nonnegative integer, let \(v^{(m)}_{d,r} (K)\) denote the least integer (if such an integer exists) with the property that whenever \(s>v_{d,r}^{(m)} (K)\), and \(f_j({\mathbf x}) \in K[x_1, \dots, x_s]\) \((i\leq j\leq r)\) are forms of degree \(d\), then the system of equations \(f_j({\mathbf x}) =0\) \((1\leq j\leq r)\) possesses a solution set which contains a \(K\)-rational linear space of projective dimension \(m\). If no such integer exists, define \(v_{d,r}^{(m)} (K)\) to be \(+\infty\). Define \(\varphi_{d,r} (K)\) in a like manner such that the arbitrary forms of degree \(d\) are restricted to be diagonal. In this paper the author obtains
(i) \(v_{3,r}^{(m)} (K)\leq r^3(m+1)^5 (\varphi_{3,r} (K)+1)^5\) if \(\varphi_{3,r} (K)\) is finite and, in particular, \[ v_{3,r}^{(m)} (\mathbb{Q})< (90r)^8 \bigl(\log (27r) \bigr)^5 (m+1)^5; \] (ii) \(v_{5,r}^{(m)} (\mathbb{Q})< \exp (10^{32} \{(m+1) r\log(3r)\}^k \log(3r (m+1))) \), where \(k=\log (3430)/ \log(4)= 5.87199, \cdots\), and, in particular, \[ v^{(0)}_{5,r} (\mathbb{Q}) =o \bigl( \exp (r^6) \bigr). \] These results deal with the quantitative part of a result by B. J. Birch [Mathematika 4, 102-105 (1957; Zbl 0081.04501)] and provide a more effective method than W. M. Schmidt’s highly developed version of the Hardy-Littlewood method [Proc. Int. Congr. Math., Warszaw 1983, Vol. 1, 515-524 (1984; Zbl 0567.10013) and Acta Math. 154, 243-296 (1985; Zbl 0561.10010)]. The author has also considered local fields in [Compos. Math 111, No. 2, 149-165 (1998; Zbl 0892.11011)].
For the entire collection see [Zbl 0874.00035].

MSC:

11E76 Forms of degree higher than two
11D72 Diophantine equations in many variables
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