Small independent zeros of quadratic forms

Let Q(x) = ∑<SUP>n</SUP><SUB>f-1</SUB> ∑<SUP>n</SUP><SUB>f-1</SUB> q<SUB>f5</SUB> x<SUB>i</SUB>x<SUB>i</SUB> 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<SUP>n</SUP> 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<SUP>n</SUP> of the equation Q(x) = 0; more precisely, if ||x|| : = max <SUB>1≤ i ≤ n</SUB> |x<SUB>i</SUB>| and ||Q|| : = max<SUB>i, f</SUB> |q<SUB>if</SUB>|, then there exists x ≠ 0 in Z<SUB>n</SUB> such that Q(x) = 0 and further ||x|| ≤ k||Q||<SUP>(n-1)/2</SUP>. (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