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- Title
Zeros of systems of p-adic quadratic forms.
- Authors
Heath-Brown, D. R.
- Abstract
We show that a system of r quadratic forms over a p-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.
- Subjects
P-adic fields; QUADRATIC forms; MATHEMATICAL variables; DIOPHANTINE equations; ALGEBRAIC geometry
- Publication
Compositio Mathematica, 2010, Vol 146, Issue 2, p271
- ISSN
0010-437X
- Publication type
Article
- DOI
10.1112/S0010437X09004497