Small independent zeros of quadratic forms

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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Let Q(x) = ∑nf-1nf-1 qf5 xixi 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 Zn 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 Zn of the equation Q(x) = 0; more precisely, if ||x|| : = max 1≤ i ≤ n |xi| and ||Q|| : = maxi, f |qif|, then there exists x ≠ 0 in Zn 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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