Defining integrality at prime sets of high density in number fields.
Let \(M\) be a totally real number field containing a subfield \(F\) of degree \(m\) over the rationals. The author proves that there is a set of primes \(W\) of Dirichlet density close to \((m-1)/m\) such that the ring of integers of \(M\) has a Diophantine definition over the ring of \(W\)-integers of \(M\) (elements of \(M\) with denominators divisible only by primes of \(W\)). She also gives a relevant estimate for the density of the primes of \(F\) below some (guaranteed to exist) \(W'\) that satisfies the same hypothesis. This improves earlier results of this type. These types of problems constitute analogues of Hilbert's Tenth Problem. The outstanding question is whether the Diophantine (equivalently, positive existential) theory of a number field is undecidable.
scientific article; zbMATH DE number 1425267
scientific article