The u-invariant of p-adic function fields

Abstract

Abstract Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.