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The implicit function theorem and free algebraic sets


Authors: Jim Agler and John E. M$^{\mathrm{c}}$Carthy
Journal: Trans. Amer. Math. Soc. 368 (2016), 3157-3175
MSC (2010): Primary 14M99, 16S50
DOI: https://doi.org/10.1090/tran/6546
Published electronically: July 22, 2015
MathSciNet review: 3451873
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Abstract | References | Similar Articles | Additional Information

Abstract: We prove an implicit function theorem for non-commutative functions. We use this to show that if $ p(X,Y)$ is a generic non-commuting polynomial in two variables and $ X$ is a generic matrix, then all solutions $ Y$ of $ p(X,Y)=0$ will commute with $ X$.


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Additional Information

Jim Agler
Affiliation: Department of Mathematics, University of California San Diego, La Jolla, California 92093

John E. M$^{\mathrm{c}}$Carthy
Affiliation: Department of Mathematics, Washington University, St. Louis, Missouri 63130

DOI: https://doi.org/10.1090/tran/6546
Keywords: NC functions, free holomorphic functions, free algebraic sets
Received by editor(s): February 19, 2014
Published electronically: July 22, 2015
Additional Notes: The first author was partially supported by National Science Foundation Grant DMS 1068830
The second author was partially supported by National Science Foundation Grant DMS 1300280
Article copyright: © Copyright 2015 American Mathematical Society

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