  In this work we perform a numerical study of a rotating, harmonically
trapped, Bose-Einstein condensate of microcavity polaritons. An efficient
numerical method (toolbox) to solve the complex Gross-Pitaevskii equation is
developed. Using this method, we investigate how the behavior of the number of
vortices formed inside the condensate changes as the various system parameters
are varied. In contrast to the atomic condensates, we show, there exists an
(experimentally realizable) range of parameter values in which all the vortices
can be made to vanish even when there is a high rotation. We further explore
how this region can be tuned through other free parameters and also discuss how
this study can help to realize the synthetic magnetic field for polaritons and
hence paving the way for the realization of the quantum Hall physics and many
other exotic phenomena.
