Heights and Quadratic Forms: Cassels’ Theorem and its Generalizations

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Mathematics (CMC)

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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.


Please note that this article was first published in Contemporary Mathematics 587 (2013) by the American Mathematical Society.

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© 2013 American Mathematical Society