Joint Logic and Number Theory Seminar

Margaret Thomas, University of Konstanz

"Effective Pila--Wilkie bounds for restricted Pfaffian surfaces"

The counting theorem of Pila and Wilkie opened up one of the most important developments in applied model theory in recent years. It provides a bound on the density of rational points of bounded height lying on the `transcendental parts' of sets definable in o-minimal expansions of the real field, a result which has had several stunning number theoretic applications (e.g. to the Manin-Mumford and André-Oort Conjectures). However, the proof of the theorem is not effective: it does not give a procedure which, given a definable set, will compute the Pila-Wilkie bound for that set. This of course constrains the effectivity of its applications. I will discuss some recent progress made towards finding an effective version of the Pila-Wilkie Theorem in certain cases. (Joint work with G.O. Jones.)

MC 5403