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SOLUBILITY OF SYSTEMS OF QUADRATIC FORMS

Published online by Cambridge University Press:  01 July 1997

GREG MARTIN
GREG MARTIN
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1003, USA
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Abstract

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.

Type
Research Article
Information
Bulletin of the London Mathematical Society , Volume 29 , Issue 4 , July 1997 , pp. 385 - 388
Copyright
© The London Mathematical Society 1997

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