Zeros of systems of p-adic quadratic forms
We show that a system of r quadratic forms over a -adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer.J.Math.87 (1...
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| Format: | Journal article |
| Language: | English |
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2010
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| Summary: | We show that a system of r quadratic forms over a -adic field in at least 4r+1 variables will have a non-trivial zero as soon as the cardinality of the residue field is large enough. In contrast, the Ax-Kochen theorem [J. Ax and S. Kochen, Diophantine problems over local fields. I, Amer.J.Math.87 (1965), 605-630] requires the characteristic to be large in terms of the degree of the field over ℚp. The proofs use a p-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry. © 2010 Foundation Compositio Mathematica. |
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