Small independent zeros of quadratic forms

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.