Heights and Quadratic Forms: Cassels’ Theorem and its Generalizations
In this survey paper, we discuss the classical Cassels' theorem on existence of small-height zeros of quadratic forms over Q and its many extensions, to different fields and rings, as well as to more general situations, such as existence of totally isotropic small-height subspaces. We also discuss related recent results on effective structural theorems for quadratic spaces, as well as Cassels'-type theorems for small-height zeros of quadratic forms with additional conditions. We conclude with a selection of open problems.
© 2013 American Mathematical Society
Fukshansky, Lenny. "Heights and Quadratic Forms: Cassels’ Theorem and its Generalizations." Diophantine Methods, Lattices, and Arithmetic Theory of Quadratic Forms. Contemporary Mathematics 587 (2013): 77-94.