A universal first-order formula defining the ring of integers in a number field (Q2510101)

The main motivation for the author comes from a natural extension of Hilbert's tenth problem to a commutative and countable ring \(R\), whose elements admit a fixed representation by integers. More precisely, for such a ring \(R\), it is asked whether there exists an algorithm that decides about the existence of solutions in \(R\) of polynomial equations with coefficients in \(R\). When \(R\) is the ring of integers of a number field, the question is an open problem, although it has been answered in the negative in many cases. When \(R\) is a number field, this becomes a classical and difficult problem in algebraic geometry, namely that of finding rational points on varieties. A connection between these two settings would come from an existential definition of the ring of integers \(\mathcal O_K\) in the number field \(K\). Unfortunately such an existential definition is still out of reach. The main result of the paper under review is a first-order universal formula defining \(\mathcal O_K\) in \(K\), which generalizes an analogous result for \(\mathbb Z\) in \(\mathbb Q\), obtained by \textit{J. Koenigsmann} [``Defining \(\mathbb Z\) in \(\mathbb Q\)'', Preprint, \url{arXiv:1011.3424}]. The proof relies on ideas of \textit{B. Poonen} [Am. J. Math. 131, No. 3, 675--682 (2009; Zbl 1179.11047)] and Koenigsmann [loc. cit], using some class field theory.