Diophantine Definability and Decidability in Large Subrings of Totally Real Number Fields and Their Totally Complex Extensions of Degree 2 Original Research Article

Let M be a number field. Let W be a set of non-archimedean primes of M. LetOM,W={xset membership, variantMmidordpxgreater-or-equal, slanted0 for allpnegated set membershipW}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper, she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved: (1) Let M be a totally real field or a totally complex extension of degree 2 of a totally real field. Then, for any var epsilon>0, there exists a set WM of primes of M whose density is greater than 1−[M:image]−1−var epsilon and such that image has a Diophantine definition over OM,WM. (Thus, Hilbertʹs Tenth Problem is undecidable in OM,WM.) (2) Let M be as above and let var epsilon>0 be given. Let Simage be the set of all rational primes splitting in M. (If the extension is Galois but not cyclic, Simage contains all the rational primes.) Then there exists a set of M-primes WM such that the set of rational primes Wimage below WM differs from Simage by a set contained in a set of density less than var epsilon and such that image has a Diophantine definition over OM,WM. (Again this will imply that Hilbertʹs Tenth Problem is undecidable in OM,WM.)