Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial
f
f
in
d
d
freely noncommuting arguments, find a free polynomial
p
n
p_n
, of degree at most
n
n
, to minimize
c
n
≔
‖
p
n
f
−
1
‖
2
c_n ≔\|p_nf-1\|^2
. (Here the norm is the
ℓ
2
\ell ^2
norm on coefficients.) We show that
c
n
→
0
c_n\to 0
if and only if
f
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
d
-shift.