2018/09/24 by Meric L. Augat, Meric Augat · 1 citation
Mathematics · #Advanced Differential Equations and Dynamical Systems #Advanced Topics in Algebra #Mathematics and Applications
paper · doi:10.1112/plms.12200
The main result of this paper establishes the free analog of Grothendieck's theorem on bijective polynomial mappings of 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-tuples 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.