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



Kirsten Eisenträger (Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, U.S.A.)

Graham Everest (School of Mathematics, University of East Anglia, Norwich NR4 7TJ, U.K.)

Alexandra Shlapentokh (Department of Mathematics, East Carolina University, Greenville, NC 27858, U.S.A.)


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 $K$.

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