Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields

Details Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields

2010-12-22

arxiv.org/abs/1012.4878v2

Mathematics

Logic

21 pages; strengthened main theorems: proved main theorems for sets of nonnegative computable real densities; minor correction

Scientific paper

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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