  We study the dynamics of roots of f'(z,t), where f(z,t) is a locally
univalent polynomial solution of the Polubarinova-Galin equation for the
evolution of the conformal map onto a Hele-Shaw blob subject to injection at
one point. We give examples of the sometimes complicated motion of roots, but
show also that the asymptotic behavior is simple. More generally we allow
f'(z,t) to be a rational function and give sharp estimates for the motion of
poles and for the decay of the Taylor coefficients. We also prove that any
global in time locally univalent solution actually has to be univalent.
