SMALL ZEROS OF QUADRATIC FORMS OVER Q

Let N ≥ 2 be an integer, F a quadratic form in N variables over Q, and Z ⊆ Q N an L-dimensional subspace, 1 ≤ L ≤ N . We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space (Z, F ). This provides an analogue over Q of a well-known theorem of Vaaler proved over number fields. We use our result to prove an effective version of Witt decomposition for a bilinear space over Q. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over Q. This extends previous results of the author over number fields. All bounds on height are explicit.