Zeros of systems of ?-adic quadratic forms
Zeros of systems of ?-adic quadratic forms
复制标题
?-adic 二次型系统的零点
DOI：
10.1112/s0010437x09004497
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发表时间：
2010
影响因子：
1.8
通讯作者：
D. R. Heath
中科院分区：
文献类型：
--
作者：
D. R. Heath
本文证明了环上r个二次型的系统，只要剩余域的基数足够大，至少有4 r +1个变量的进域将有一个非平凡的零。相反，Ax-Kochen定理[J. AxandS.柯亨，局部域上的丢番图问题。I，Amer.J.Math.87（1965），605-630]要求该特征在场在Pwp上的程度方面大。证据是用一个？adic最小化技术，以及剩余类域上的计数参数，基于代数几何的考虑。
Abstract 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 ?-adic minimization technique, together with counting arguments over the residue class field, based on considerations from algebraic geometry.