# Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation

## TL;DR

This paper develops a local theory for noncommutative functions, establishing properties of germs, meromorphic functions, and interpolation, revealing new algebraic structures and phenomena in free analysis.

## Contribution

It introduces the ring of noncommutative germs at scalar points as an integral domain with a skew field of fractions and proves a free Hermite interpolation theorem for noncommutative functions.

## Key findings

- The ring of germs at scalar points is an integral domain with a universal skew field of fractions.
- Existence of nonzero nilpotent functions near semisimple matrices shows complex local structures.
- A free Hermite interpolation theorem guarantees polynomial approximation of noncommutative functions at multiple points.

## Abstract

Free analysis is a quantization of the usual function theory much like operator space theory is a quantization of classical functional analysis. Basic objects of free analysis are noncommutative functions. These are maps on tuples of matrices of all sizes that preserve direct sums and similarities. This paper investigates the local theory of noncommutative functions. The first main result shows that for a scalar point $Y$, the ring $O_Y$ of uniformly analytic noncommutative germs about $Y$ is an integral domain and admits a universal skew field of fractions, whose elements are called meromorphic germs. A corollary is a local-global rank principle that connects ranks of matrix evaluations of a matrix $A$ over $O_Y$ with the factorization of $A$ over $O_Y$. Different phenomena occur for a semisimple tuple of non-scalar matrices $Y$. Here it is shown that $O_Y$ contains copies of the matrix algebra generated by $Y$. In particular, there exist nonzero nilpotent uniformly analytic functions defined in a neighborhood of $Y$, and $O_Y$ does not embed into a skew field. Nevertheless, the ring $O_Y$ is described as the completion of a free algebra with respect to the vanishing ideal at $Y$. This is a consequence of the second main result, a free Hermite interpolation theorem: if $f$ is a noncommutative function, then for any finite set of semisimple points and a natural number $L$ there exists a noncommutative polynomial that agrees with $f$ at the chosen points up to differentials of order $L$. All the obtained results also have analogs for (non-uniformly) analytic germs and formal germs.