Linear Spaces on the Intersection of Two Quadratic Hypersurfaces,and Systems of p-adic Quadratic Forms

We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field . One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over . As a corollary we also get new bounds for the number of variables necessary to always find a non-trivial p-adic zero of a system of quadratic forms.