Polynomial Gaussian beams, which are laser beams with Gaussian envelops and complex bivariate polynomial
prefactors, provide us with a tractable means to investigate the evolution of optical vortices. A formalism for
the propagation of such beams allows one to determine how the coefficients of the polynomial transform during
propagation. This formalism is used to proof that global topological charge is conserved, provided that the
Gaussian envelope of the beam is rotationally symmetric. For astigmatic Gaussian beams the global topological
charge is not conserved and can change during propagation in steps of 2 when one of the optical vortices undergoes
topological charge inversion. The global topological charge is bounded by the order of the polynomial prefactor.
One can also investigate the behavior of vortices in random vortex fields by modelling them as polynomial
Gaussian beams. The phase functions that exist in the vicinity of the annihilation of a vortex dipole are similar,
regardless of the type of beam in which the vortices exist. One can therefore use polynomial Gaussian beams to
find a way to force vortices in random vortex fields to annihilate. The number of optical vortices that can exist
in a polynomial Gaussian beam depends on the reducibility of the polynomial prefactor. During propagation
the reducibility of the prefactor is generally destroyed. However, if the morphologies of the vortices of a fully
reducible prefactor are all the same, the reducibility is maintained during propagation. The results obtained
from the analyses of polynomial Gaussian beams are confirmed by numerical simulations.