# Hilbert’s Tenth Problem inside the rationals

## Russell Miller

### City University of New York

For a ring *R*, Hilbert’s Tenth Problem is the set *HTP(R)* of polynomials *f ∈ R[X _{1},X_{2},…]* for which

*f=0*has a solution in

*R*. Matiyasevich, completing work of Davis, Putnam, and Robinson, showed that

*HTP(*is Turing-equivalent to the Halting Problem. The Turing degree of

**Z**)*HTP(*remains unknown. Here we consider the problem for subrings of

**Q**)*. One places a natural topology on the space of such subrings, which is homeomorphic to Cantor space. This allows consideration of measure theory and also Baire category theory. We prove, among other things, that*

**Q***HTP(*computes the Halting Problem if and only if

**Q**)*HTP(R)*computes it for a nonmeager set of subrings

*R*.

Russell Miller is professor of mathematics at Queens College of CUNY and also at the CUNY Graduate Center. He conducts research in mathematical logic, especially computability theory and its interaction with other areas of mathematics, as in computable model theory. He received his doctorate from the University of Chicago in 2000, as a student of Robert Soare, and subsequently held a postdoctoral position at Cornell University until 2003, when he came to CUNY.