Baire Category Theory and Hilbert’s Tenth Problem Inside $\mathbb {Q}$

Abstract

For a ring R, Hilbert’s Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP$(\mathbb {Q})$, whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP$(\mathbb {Q})$ computes C if and only if the subrings R for which HTP(R) computes C form a nonmeager class. Similar results hold for 1-reducibility, for admitting a Diophantine model of $\mathbb {Z}$, and for existential definability of $\mathbb {Z}$.

R. Miller—The author was supported by Grant # DMS – 1362206 from the N.S.F., and by several grants from the PSC-CUNY Research Award Program. This work grew out of research initiated at a workshop at the American Institute of Mathematics and continued at a workshop at the Institute for Mathematical Sciences of the National University of Singapore. Conversations with Bjorn Poonen and Alexandra Shlapentokh have been very helpful in the creation of this article.