The inverse function theorem and the Jacobian conjecture for free analysis

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.Question 1.2. Let P : C N → C N be a polynomial map. If the Jacobian DP (x) is invertible for every x ∈ C N , is the map P itself invertible?

Aspects of non-commutative function theory

“…Pascoe proved the following inverse function theorem in [24]. The equivalence of (i) and (iii) is due to Helton, Klep and McCullough [14].…”

“…Since this is the largest admissible topology, for any admissible topology , any -holomorphic function is automatically fine holomorphic. The class of nc functions considered in [14,24] is the fine holomorphic functions.…”

The free Grothendieck theorem

“…Of course if is injective, then considered as a polynomial in commuting variables, it is bijective and has a polynomial inverse. The following free polynomial analog of Grothendieck's theorem was implicitly conjectured in . Theorem If is an injective free polynomial mapping, then there is a free polynomial mapping such that ; that is, has a free polynomial inverse.…”

“…Subsection 4.1 explores conditions that guarantee a free polynomial has a free polynomial inverse. While in Subsection 4.2 we recall the free derivative as defined in and investigate its properties. For a fixed free polynomial , we define the function and observe that Pascoe's solution to the free Jacobian conjecture can be interpreted as saying, is bijective if and only if is bijective.…”

Non-commutative holomorphic functions on operator domains

“…[7,8,9,10,11,12,15,16]. Kaliuzhnyi-Verbovetskyi and Vinnikov have written a monograph [13] which develops the important ideas of the subject.…”