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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

   
 
 

 

The classical Artin approximation theorems


Author: Herwig Hauser
Journal: Bull. Amer. Math. Soc. 54 (2017), 595-633
MSC (2010): Primary 13-02, 14-02, 32-02
DOI: https://doi.org/10.1090/bull/1579
Published electronically: June 13, 2017
Original version: Posted June 13, 2017.
Corrected version: Current version includes addition of the word "uncountable" on page 9.
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Abstract: The various Artin approximation theorems assert the existence of power series solutions of a certain quality $ Q$ (i.e., formal, analytic, algebraic) of systems of equations of the same quality $ Q$, assuming the existence of power series solutions of a weaker quality $ Q' < Q$ (i.e., approximated, formal). The results are frequently used in commutative algebra and algebraic geometry. We present a systematic argument which proves, with minor modifications, all theorems simultaneously. More involved results, such as, e.g., Popescu's nested approximation theorem for algebraic equations or statements about the Artin function, will only be mentioned but not proven. We complement the article with a brief account of the theory of algebraic power series, two applications of approximation to singularities, and a differential-geometric interpretation of Artin's proof.


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Additional Information

Herwig Hauser
Affiliation: Faculty of Mathematics, University of Vienna, Austria
Email: herwig.hauser@univie.ac.at

DOI: https://doi.org/10.1090/bull/1579
Received by editor(s): December 2, 2016
Published electronically: June 13, 2017
Additional Notes: Supported by the Austrian Science Fund FWF within the projects P25652 and AI0038211
Dedicated: To Michael Artin
Article copyright: © Copyright 2017 by the author