Small zeros of quadratic forms outside of a union of varieties

Abstract: Let $F$ be a quadratic form in $N \geq 2$ variables defined on a vector space $V \subseteq K^N$ over a global field $K$, and $\Z \subseteq K^N$ be a finite union of varieties defined by families of homogeneous polynomials over $K$. We show that if $V \setminus \Z$ contains a nontrivial zero of $F$, then there exists a linearly independent collection of small-height zeros of $F$ in $V\setminus \Z$, where the height bound does not depend on the height of $\Z$, only on the degrees of its defining polynomials. As a corollary of this result, we show that there exists a small-height maximal totally isotropic subspace $W$ of the quadratic space $(V,F)$ such that $W$ is not contained in $\Z$. Our investigation extends previous results on small zeros of quadratic forms, including Cassels' theorem and its various generalizations. The paper also contains an appendix with two variations of Siegel's lemma. All bounds on height are explicit.