Seminars & Colloquia Calendar
Hilbert's Tenth Problem for Subrings of the Rationals
Russell Miller (Queens College)
Location: Hill 705
Date & time: Monday, 29 January 2018 at 5:00PM - 6:00PM
Abstract: For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of all polynomials $fin R[X_1,X_2,ldots]$ 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, $HTP(mathbb{Z})$, is undecidable. On the other hand, the decidability of $HTP(mathbb{Q})$ 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 $HTP(mathbb{Q})$ itself.
Some of the work discussed is joint with Kramer, and some with Eisentraeger, Park, and Shlapentokh.
