Problems Connecting Logic and Number Theory

Bjorn Poonen discussed the “recognition problem” for finitely generated rings (and fields). That is, given two finitely generated commutative rings A and B, presented in terms of generators and relations, is there a decision procedure to determine whether or not these rings are isomorphic (this being, one would think, a basic issue for algebraic geometry!). Of course, if one drops the requirement of commutativity, one comes up against the unsolvablity of the corresponding problem for finitely generated groups (by taking A and B simply to be integral group rings). Carol Wood brought up cases where model theory, applied to number theoretic problems provided bounds that are impressively good! Model Theory—in some instances—yields significantly new proofs of theorems obtained by the number-theorists . In other instances, model theory achieves startling results for problems not yet considered by number theorists . Carol Wood discussed the recent article of Pila and Wilkie ([PW07]) that provides asymptotic upper bounds (as a function of the variable T ) for the number of Q-rational points of height ≤ T that