  A polynomial-in-time growth bound is established for global Sobolev
$H^s(\mathbb T)$ solutions to the derivative nonlinear Schr\"odinger equation
on the circle with $s>1$. These bounds are derived as a consequence of a
nonlinear smoothing effect for an appropriate gauge-transformed version of the
periodic Cauchy problem, according to which a solution with its linear part
removed possesses higher spatial regularity than the initial datum associated
with that solution.
