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 $X$.

References in corpus (2)

• Proper Analytic Free Maps
• Analytic mappings between noncommutative pencil balls