Model theory seminar - Travis Morrison
Speaker: Travis Morrison, Waterloo
Title: Definability and decidability in global fields
Hilbert's tenth problem asks for an algorithm which takes as input a polynomial with integer coefficients and decides whether or not that polynomial has a root with integer coordinates. The work of Davis-Putnam-Robinson and Matiyasevich shows that no such algorithm exists. Hilbert's tenth problem for the rationals is open: it is not known if there can exist an algorithm which decides whether a rational polynomial has a zero in the rationals. If the integers are a diophantine subset of the rationals, meaning if there is a first-order definition of the integers as a subset of the rationals (in the language of rings) involving only existential quantifiers, then Hilbert's tenth problem for the rationals would be undecidable. Such a definition may not exist, but work of Koenigsmann shows there is a universal definition. I will discuss this result and its generalizations to number fields, due to Park, and to global function fields, which is joint work with Eisentraeger. These definitions are built using ideas from class field theory, local-global principles, and the arithmetic of quaternion algebras.