The implicit function theorem and free algebraic sets

TL;DR

This paper establishes an implicit function theorem for non-commutative functions and demonstrates that solutions to certain polynomial equations involving generic matrices commute with the given matrices.

Contribution

It introduces a non-commutative implicit function theorem and applies it to show commutativity of solutions for generic polynomial equations.

Findings

Solutions to generic polynomial equations commute with the given matrix

The implicit function theorem extends to non-commutative functions

Generic polynomials in non-commuting variables have solutions with specific algebraic properties

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 $X$.