Zeros of systems of 𝔭-adic quadratic forms

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 ℚ_p. The proofs use a 𝔭-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.