The inverse function theorem and the resolution of the Jacobian conjecture in free analysis

TL;DR

This paper proves an invertibility criterion for free polynomials and functions, leading to a resolution of the Jacobian conjecture in free analysis and for commuting matrices, by linking derivative nonsingularity to invertibility.

Contribution

It introduces a new invertibility criterion for free functions and polynomials, and applies it to resolve the Jacobian conjecture within free and commutative matrix contexts.

Findings

Invertibility criterion for free polynomials and functions

Resolution of the Jacobian conjecture in free analysis

Applicability to commuting matrices

Abstract

We establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices. We show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity, the function must be invertible. Thus, as a corollary, we establish the Jacobian conjecture in this context. Furthermore, our result holds for commutative polynomials evaluated on tuples of commuting matrices.