The free Grothendieck theorem

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.

40 pages

References in corpus (2)

• Proper Analytic Free Maps
• Regular and positive noncommutative rational functions

Cited by in corpus (4)

• Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
• Noncommutative free universal monodromy, pluriharmonic conjugates, and plurisubharmonicity
• Free potential functions
• Free noncommutative principal divisors and commutativity of the tracial fundamental group