Small zeros of quadratic forms over the algebraic closure of Q

Abstract

Let $N \geq 2$ 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, $1 \leq L \leq N$. 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.