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Preprint Number 1006
1006. Russell Miller
Baire category theory and Hilbert's Tenth Problem inside Q
Submission date: 10 February 2016.
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.
Mathematics Subject Classification: 11U05 (Primary) 03D45, 12L05 (Secondary)
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