  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.
