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Essential dimension


Author: Alexander S. Merkurjev
Journal: Bull. Amer. Math. Soc. 54 (2017), 635-661
MSC (2010): Primary 14L30, 20G10, 11E72
DOI: https://doi.org/10.1090/bull/1564
Published electronically: December 19, 2016
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Abstract: In this paper we survey research on the essential dimension that was introduced by J. Buhler and Z. Reichstein. Informally speaking, the essential dimension of a class of algebraic objects is the minimal number of algebraically independent parameters one needs to define any object in the class. The notion of essential dimension, which is defined in elementary terms, has surprising connections to many areas of algebra, such as algebraic geometry, algebraic $ K$-theory, Galois cohomology, representation theory of algebraic groups, theory of fibered categories and valuation theory. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups.


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Additional Information

Alexander S. Merkurjev
Affiliation: Department of Mathematics, University of California at Los Angeles, Los Angeles, California 90095-1555
Email: merkurev@math.ucla.edu

DOI: https://doi.org/10.1090/bull/1564
Received by editor(s): October 17, 2016
Published electronically: December 19, 2016
Additional Notes: The work has been supported by the NSF grant DMS #1160206
Article copyright: © Copyright 2016 American Mathematical Society

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