The implicit function theorem and free algebraic sets

Jim Agler; John E. McCarthy

doi:10.1090/tran/6546

The implicit function theorem and free algebraic sets

Jim Agler
, John E. McCarthy

Research output: Contribution to journal › Article › peer-review

23 Scopus citations

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.

Original language	English
Pages (from-to)	3157-3175
Number of pages	19
Journal	Transactions of the American Mathematical Society
Volume	368
Issue number	5
DOIs	https://doi.org/10.1090/tran/6546
State	Published - May 2016

Keywords

Free algebraic sets
Free holomorphic functions
NC functions

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10.1090/tran/6546

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title = "The implicit function theorem and free algebraic sets",

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.",

keywords = "Free algebraic sets, Free holomorphic functions, NC functions",

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AB - 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.

KW - Free algebraic sets

KW - Free holomorphic functions

KW - NC functions

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JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

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The implicit function theorem and free algebraic sets

Abstract

Keywords

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