Solubility of Systems of Quadratic Forms

It has been known since the last century that a single quadratic form in at least five variables has a nontrivial zero in any p ‐adic field, but the analogous question for systems of quadratic forms remains unanswered. It is plausible that the number of variables required for solubility of a system of quadratic forms simply is proportional to the number of forms; however, the best result to date, from an elementary argument of Leep [ 6 ], is that the number of variables needed is at most a quadratic function of the number of forms. The purpose of this paper is to show how these elementary arguments can be used, in a certain class of fields including the p ‐adic fields, to refine the upper bound for the number of variables needed to guarantee solubility of systems of quadratic forms. This result partially addresses Problem 6 of Lewis' survey article [ 7 ] on Diophantine problems. 1991 Mathematics Subject Classification 11D72.