The free Grothendieck theorem

TL;DR

This paper proves a free analog of Grothendieck's theorem, showing that injective free polynomial mappings of matrices have polynomial inverses, and introduces an algorithm to test invertibility, along with a new class of free algebraic functions.

Contribution

It establishes the free Grothendieck theorem for matrix-valued polynomial mappings and introduces hyporational functions, expanding the understanding of free algebraic structures.

Findings

Injective free polynomial mappings have polynomial inverses.

An algorithm to test invertibility of free polynomial mappings.

Identification of hyporational functions between free rational and algebraic functions.

Abstract

The main result of this article establishes the free analog of Grothendieck's Theorem on bijective polynomial mappings of $\mathbb{C}^g$. Namely, we show if $p$ is a polynomial mapping in $g$ freely non-commuting variables sending $g$-tuples of matrices (of the same size) to $g$-tuple of matrices (of the same size) that is injective, then it has a free polynomial inverse. Other results include an algorithm that tests if a free polynomial mapping $p$ has a polynomial inverse (equivalently is injective; equivalently is bijective). Further, a class of free algebraic functions, called hyporational, lying strictly between the free rational functions and the free algebraic functions are identified. They play a significant role in the proof of the main result.