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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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The classical Artin approximation theorems
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by Herwig Hauser PDF
Bull. Amer. Math. Soc. 54 (2017), 595-633

Corrigendum: Bull. Amer. Math. Soc. 55 (2018), 289-293.
Original version: Posted June 13, 2017.
Corrected version: Current version includes addition of the word "uncountable" on page 9.


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
  • MR Author ID: 82620
  • Email:
  • 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
  • © Copyright 2017 by the author
  • Journal: Bull. Amer. Math. Soc. 54 (2017), 595-633
  • MSC (2010): Primary 13-02, 14-02, 32-02
  • DOI:
  • MathSciNet review: 3683627