Zeros of systems of p-adic quadratic forms

Journal article

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.

Authors: Heath-Brown, D

• Publication status: Published
• Journal: COMPOSITIO MATHEMATICA
• Volume: 146
• Issue: 2
• Pages: 271-287
• Publication date: 2010-03-01
• DOI: 10.1112/S0010437X09004497
• EISSN: 1570-5846
• ISSN: 0010-437X
• Language: English
• Keywords: p-adic field; system; zeros; quadratic form; Ax-Kochen theorem