  We derive an upper bound for the least number of variables needed to
guarantee that a system of t quadratic forms (t>=2) over a field F has a
nontrivial zero. In particular, if F is a local field, then 2t^2+3 variables
insure the existence of a nontrivial zero (2t^2+1 if t is even), while if F=Q_p
with p>=11, then 2t^2-2t+5 variables suffice (2t^2-2t+1 if 3 divides t). The
improvement lies in a more efficient use of information on the solubility of
pairs and triplets of quadratic forms, and the arguments are completely
elementary.
