Totally isotropic subspaces of small height in quadratic spaces

TL;DR

This paper proves the existence of infinitely many small-height maximal totally isotropic subspaces in quadratic spaces over global fields, extending Vaaler's theorem and providing explicit height bounds.

Contribution

It generalizes Vaaler's theorem by constructing infinite families of small-height isotropic subspaces that span the space, with explicit height bounds.

Findings

Existence of infinite families of small-height isotropic subspaces.

Explicit bounds on the height of these subspaces.

Extension of Vaaler's theorem to broader settings.

Abstract

Let $K$ be a global field or $\overline{\mathbb Q}$, $F$ a nonzero quadratic form on $K^N$, $N \geq 2$, and $V$ a subspace of $K^N$. We prove the existence of an infinite collection of finite families of small-height maximal totally isotropic subspaces of $(V,F)$ such that each such family spans $V$ as a $K$-vector space. This result generalizes and extends a well known theorem of J. Vaaler and further contributes to the effective study of quadratic forms via height in the general spirit of Cassels' theorem on small zeros of quadratic forms. All bounds on height are explicit.