Wilkie's conjecture for restricted elementary functions

摘要

We consider the structure R-RE obtained from (R, <, +, ") by adjoining the restricted exponential and sine functions. We prove Wilkie's conjecture for sets definable in this structure: the number of rational points of height H in the transcendental part of any definable set is bounded by a polynomial in log H. We also prove two refined conjectures due to Pila concerning the density of algebraic points from a fixed number field, or with a fixed algebraic degree, for RE-definable sets.