Small zeros of quadratic forms over the algebraic closure of Q

Let $N\ge 2$ be an integer, $F$ a quadratic form in $N$ variables over $\bar{\mathbb{Q}}$, and $Z\subseteq {\bar{\mathbb{Q}}}^N$ an $L$-dimensional subspace, $1\le L\le N$. We prove the existence of a small-height maximal totally isotropic subspace of the bilinear space $(Z,F)$. This provides an analogue over $\bar{\mathbb{Q}}$ 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 $\bar{\mathbb{Q}}$. We also include some related effective results on orthogonal decomposition and structure of isometries for a bilinear space over $\bar{\mathbb{Q}}$. This extends previous results of the author over number fields. All bounds on height are explicit.