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ON ARTIN'S CONJECTURE, I: SYSTEMS OF DIAGONAL FORMS

Published online by Cambridge University Press:  01 May 1999

J. BRÜDERN
Affiliation:
Mathematisches Institut A, Universität Stuttgart, D-70550 Stuttgart, Germany
H. GODINHO
Affiliation:
Departimento de Matematica, Universidade de Brasilia, Brasilia, DF 70910-900, Brazil
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Abstract

As a special case of a well-known conjecture of Artin, it is expected that a system of R additive forms of degree k, say

formula here

with integer coefficients aij, has a non-trivial solution in ℚp for all primes p whenever

formula here

Here we adopt the convention that a solution of (1) is non-trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p-adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions

formula here

are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N[Gt ]R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints

formula here

to be sufficient for p-adic solubility of (1).

A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2.

Type
NOTES AND PAPERS
Copyright
© The London Mathematical Society 1999

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