# Seminars & Colloquia Calendar

Logic Seminar

## 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.

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