Model Theory Seminar - "Effective Pila--Wilkie bounds for Pfaffian surfaces", Margaret Thomas
Department of Mathematics and Statistics
Title: Effective Pila--Wilkie bounds for Pfaffian surfaces
Abstract: The counting theorem of Pila and Wilkie has become celebrated as one of the most important developments in o-minimality in recent years, with far-reaching consequences in diophantine geometry. It provides a `subpolynomial' bound on the number of rational points of bounded `height' lying on the `transcendental parts' of sets definable in o-minimal expansions of the real field. The proof of this theorem does not, however, provide an effective bound, and the pursuit of examples for which one can obtain this remains very active, with a view to number-theoretic applications. I will discuss some recent progress made (in joint work with Gareth O. Jones) on obtaining this in the particular setting of surfaces implicitly defined from `Pfaffian' functions, in both restricted and unrestricted settings.