Wilkie's conjecture for restricted elementary functions

Gal Binyamini; Dmitry Novikov

doi:10.4007/annals.2017.186.1.6

Wilkie's conjecture for restricted elementary functions

Gal Binyamini, Dmitry Novikov

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	237-275
Number of pages	39
Journal	Annals of Mathematics
Volume	186
Issue number	1
DOIs	https://doi.org/10.4007/annals.2017.186.1.6
State	Published - 1 Jul 2017

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

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abstract = "We consider the structure ℝRE 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 ℝRE-definable sets.",

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AB - We consider the structure ℝRE 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 ℝRE-definable sets.

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DO - 10.4007/annals.2017.186.1.6

M3 - مقالة

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EP - 275

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -

Wilkie's conjecture for restricted elementary functions

Abstract

All Science Journal Classification (ASJC) codes

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