Baire category theory and hilbert’s tenth problem inside 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(ℚ), whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP(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 ℤ, and for existential definability of ℤ.