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arXiv · 1404.6032

The implicit function theorem and free algebraic sets

Jim Agler·John E. McCarthy
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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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Jim Agler, John E. McCarthy. 2014-04-24. The implicit function theorem and free algebraic sets. https://arxiv.org/abs/1404.6032

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