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Solubility of Systems of Quadratic Forms
Author(s) -
Martin Greg
Publication year - 1997
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609396002664
Subject(s) - mathematics , diophantine equation , quadratic equation , class number , quadratic form (statistics) , solubility , quadratic field , binary quadratic form , field (mathematics) , pure mathematics , class (philosophy) , argument (complex analysis) , quadratic function , combinatorics , computer science , geometry , biochemistry , chemistry , organic chemistry , artificial intelligence
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.