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# Mathematical Research Letters

## Volume 18 (2011)

### Number 6

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

Pages: 1141 – 1162

DOI: https://dx.doi.org/10.4310/MRL.2011.v18.n6.a7

#### Authors

#### Abstract

We show that Hilbert’s Tenth Problem is undecidable for complementary subrings of number fields and that the $\pp$-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 $\delta_1,\ldots,\delta_t$ whose sum is one, we prove that the nonarchimedean primes of $K$ can be partitioned into $t$ disjoint recursive subsets $S_1, \dots, S_t$ of densities $\delta_1,\ldots,\delta_t$, 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 $\OO_{K,S_i}$ 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 $K$.

Published 20 February 2012