Cook, R. J. ; Raghavan, S.
(1987)
*Small independent zeros of quadratic forms
*
Mathematical Proceedings of the Cambridge Philosophical Society, 102
(1).
pp. 5-16.
ISSN 0305-0041

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Official URL: http://journals.cambridge.org/action/displayAbstra...

Related URL: http://dx.doi.org/10.1017/S0305004100066998

## Abstract

Let Q(x) = ∑^{n}_{f-1} ∑^{n}_{f-1} q_{f5} x_{i}x_{i} be a non-degenerate quadratic form with integral coefficients. Further, let Q(x) be a zero form, i.e. let there exist x ≠ 0 in Z^{n} such that Q(x) = 0. Then we know from Cassels[2], (Davenport[6] and 'a slightly more general result' from Birch and Davenport [1]) that there exists a 'small' solution x in Z^{n} of the equation Q(x) = 0; more precisely, if ||x|| : = max _{1≤ i ≤ n} |x_{i}| and ||Q|| : = max_{i, f} |q_{if}|, then there exists x ≠ 0 in Z_{n} such that Q(x) = 0 and further ||x|| ≤ k||Q||^{(n-1)/2}. (Here, and throughout this section, k will denote a number, not necessarily the same at each occurrence, which depends only on n.) An analogue of this estimate for 'integral' quadratic forms over algebraic number fields was proved in [8], with the exponent (n - 1)/2 remaining intact.

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Deposited On: | 12 Mar 2012 15:46 |

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