Article - postprint
Claremont McKenna College, Mathematics (CMC)
Let be a quadratic form in variables defined on a vector space over a global field , and be a finite union of varieties defined by families of homogeneous polynomials over . We show that if contains a nontrivial zero of , then there exists a linearly independent collection of small-height zeros of in , where the height bound does not depend on the height of , 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 of the quadratic space such that is not contained in . 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.
© 2014 American Mathematical Society
Chan, Wai Kiu, Lenny Fukshansky, and Glenn Henshaw. "Small zeros of quadratic forms missing a union of varieties." Transactions of the AMS, vol. 366 no. 10 (2014), 5587-5612.