Wilkie's conjecture for restricted elementary functions

• 成果类型: Article
• 署名作者: Binyamini, Gal; Novikov, Dmitry
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.186.1.6
• 发表日期: 2017
• 页码: 237-275
• 关键词: rational-points; subanalytic surface; sets; complexity; families; number

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.