Abstract
We obtain new results on linear spaces on the intersection of two quadratic forms defined over a non-dyadic p-adic field \({{\Bbb Q}_p}\). One of our main tools is a recent result of Parimala and Suresh on isotropy of quadratic forms over functions fields over \({{\Bbb Q}_p}\). 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.
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