Small Zeros of Quadratic Forms over the Algebraic Closure of Q

Let N \u3e= 2 be an integer, F a quadratic form in N variables over (Q) over bar, and Z subset of (Q) over bar (N) an L-dimensional subspace, 1 \u3c= L \u3c= 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) over bar 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) over bar. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over (Q) over bar. This extends previous results of the author over number fields. All bounds on height are explicit