Solubility of Systems of Quadratic Forms

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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 2t²+3 variables insure the existence of a nontrivial zero (2t²+1 if t is even), while if F=Qₚ with p>=11, then 2t²-2t+5 variables suffice (2t²-2t+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.