Title: Hilbert's Tenth Problem for Subrings of the Rationals
Speaker: Russell Miller (CUNY)
Abstract: For a ring R, Hilbert's Tenth Problem HTP(R) is the set of all polynomials f in R[X_1,X_2,...] for which f=0 has a solution in R. In 1970, Matiyasevich completed work by Davis, Putnam, and Robinson to show that the original Tenth Problem of Hilbert, HTP(Z), is undecidable. On the other hand, the decidability of HTP(Q) remains an open question. We will examine this problem for subrings of the rational numbers, viewing these subrings as the elements of a topological space homeomorphic to Cantor space and connecting their Turing degrees and computability-theoretic properties to those of HTP(Q) itself.
Some of the work discussed is joint with Kramer, and some with Eisentraeger, Park, and Shlapentokh.