Effective Pila--Wilkie bounds for restricted Pfaffian surfaces

Margaret Thomas (Konstanz)

Frank Adams 1,

The counting theorem of Pila and Wilkie has become celebrated for opening 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.)

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