An Optimal Approximation Problem For Free Polynomials

TL;DR

This paper investigates the approximation of noncommutative polynomials by free polynomials within the row ball domain, establishing conditions for convergence and providing bounds, with applications to polynomial cyclicity for the d-shift.

Contribution

It introduces a noncommutative approximation problem for free polynomials, characterizes when approximation converges, and offers new insights into polynomial cyclicity for the d-shift.

Findings

Convergence of approximation iff the polynomial is nonsingular in the row ball

Quantitative bounds on approximation error

New proof of polynomial cyclicity characterization for the d-shift

Abstract

Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n := \|p_nf-1\|^2$. (Here the norm is the $\ell^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.