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(0) |a| = max |ai| ≤ cnQ(n−1)/2 where cn is a constant which depends only on n and where Q = ∑ i,j |qij |. In the addendum to [1] he showed that the exponent (n − 1)/2 is best possible by giving an example which was found by M. Kneser. In his book [3, 6.8] he gave a better proof and estimate, in particular one can take cn = 3(n−1)/2. For a generalization to totally isotropic subspaces of higher dimension see [8]. Several years later Cassels [2] published his theorem on the representation of a polynomial f(x) in one variable as a sum of n squares: If such a representation is possible over the rational function field k(x) then it is already possible over the polynomial ring k[x]. He remarked that the underlying geometrical idea for the proof was essentially the same as in his first paper [1]. This idea is as follows: Given a 6= 0 with q(a) = 0, intersect the “quadric” q = 0 with a “line” l = 0 passing through a. If l is chosen carefully then the second intersection point of q and l may be “smaller” than the original point a. For the choice of l one has to use the fact that Z resp. k[x] are euclidean domains. These results have been generalized in the following directions:

Small zeros of quadratic forms over algebraic function fields