数域补子环中的Hilbert第十问题与Mazur猜想
Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields
摘要：
我们发现，希尔伯特第十问题是不可判定的互补子环的数域和p-adic和阿基米德环版本的马祖尔的代数不持有这些环。更具体地，给定数域K、正整数t>1和t个非负可计算实数delta_1，.，delta_t，其和为1，我们证明了K的非阿基米德素数可以划分为t个不相交的递归子集S_1，…，密度δ_1，...，delta_t，使得希尔伯特第十问题对于每个相应的环O_{K,S_i}是不可判定的。我们还表明，我们可以找到一个分区如上所述，使每个环O_{K,S_i}拥有一个无限丢番图集，这是离散的领域的每一个拓扑。关于K，我们需要的唯一假设是存在一条定义在K上的秩为1的椭圆曲线。
英文摘要：
We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t>1, and t nonnegative computable real numbers delta_1,..., delta_t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S_1,..., S_t of densities delta_1,..., delta_t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O_{K,S_i}. We also show that we can find a partition as above such that each ring O_{K,S_i} possesses an infinite Diophantine set which is discrete in every topology of the field. The only assumption on K we need is that there is an elliptic curve of rank one defined over K.
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