Methods and Applications of Analysis

Volume 21 (2014)

Number 4

Special issue dedicated to the 60th birthday of Stephen S.-T. Yau: Part II

Guest editors: John Erik Fornæss, Xiaojun Huang, Song-Ying Li, Yat Sun Poon, Wing Shing Wong, and Zhouping Xin

Artin’s approximation theorems and Cauchy-Riemann geometry

Pages: 481 – 502



Nordine Mir (Science Program, Texas A&M University at Qatar, Doha, Qatar)


Artin’s approximation theorems are powerful tools in analytic and algebraic geometry for finding solutions of systems of analytic or algebraic equations whenever a given formal solution exists. In this survey article we describe the recent developments involving the use of Artin’s approximation theorems in some problems arising from Cauchy-Riemann geometry. The solution to such problems simultaneously lead to a number of results that can be stated as PDE versions of Artin’s approximation theorems. The article is intended to a non-expert audience. A number of examples and open problems are also mentioned.


holomorphic map, formal map, algebraic map, CR manifold, Artin approximation

2010 Mathematics Subject Classification

14P05, 14P15, 14P20, 32C05, 32C07, 32H02, 32V05, 32V20, 32V40

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