Matrix evaluations of noncommutative rational functions and Waring problems

TL;DR

This paper demonstrates that large matrices can realize noncommutative rational functions with distinct eigenvalues and applies this to solve Waring problems, showing that matrices can be expressed as differences or products of polynomial evaluations.

Contribution

It establishes the existence of matrices with prescribed eigenvalues for noncommutative rational functions and applies this to Waring problems in matrix algebras, including polynomial and rational cases.

Findings

Existence of matrices with n distinct eigenvalues for large n

Every matrix in sl_n(K) can be expressed as a difference of rational function evaluations

Every nonscalar matrix in GL_n(K) can be written as a product of polynomial evaluations

Abstract

Let $r$ be a nonconstant noncommutative rational function in $m$ variables over an algebraically closed field $K$ of characteristic 0. We show that for $n$ large enough, there exists an $X\in M_n(K)^m$ such that $r(X)$ has $n$ distinct and nonzero eigenvalues. This result is used to study the linear and multiplicative Waring problems for matrix algebras. Concerning the linear problem, we show that for $n$ large enough, every matrix in $sl_n(K)$ can be written as $r(Y)-r(Z)$ for some $Y,Z\in M_n(K)^m$. We also discuss variations of this result for the case where $r$ is a noncommutative polynomial. Concerning the multiplicative problem, we show, among other results, that if $f$ and $g$ are nonconstant polynomials, then, for $n$ large enough, every nonscalar matrix in $GL_n(K)$ can be written as $f(Y)g(Z)$ for some $Y,Z\in M_n(K)^m$.