Free function theory through matrix invariants

TL;DR

This paper develops invariant theory tools to analyze free maps with involution, characterizing polynomial maps, establishing power series expansions, and proving inverse and implicit function theorems in the noncommutative setting.

Contribution

It introduces invariant-theoretic methods to free analysis with involution, providing new characterizations, power series expansions, and inverse/implicit function theorems for free maps.

Findings

Free maps with involution do not have strong rigidity properties.

Polynomial free maps can be characterized by properties of their finite-dimensional slices.

Power series expansions for analytic free maps are established, including at non-scalar points.

Abstract

In this article we introduce powerful tools and techniques from invariant theory to free analysis. This enables us to study free maps with involution. These maps are free noncommutative analogs of real analytic functions of several variables. With examples we demonstrate that they do not exhibit strong rigidity properties of their involution-free free counterparts. We present a characterization of polynomial free maps via properties of their finite-dimensional slices. This is used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are given by series of generalized polynomials for which we obtain an invariant-theoretic characterization. Finally, we give an inverse and implicit function theorem for free maps with involution.