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Abstract:

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 R^{RE}-definable sets.

Mathematics Subject Classification: 11G99 (Primary) 03C64 11U09 (Secondary)

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Full text arXiv 1605.04671: pdf, ps.