Minimal polynomial systems for point vortex equilibria

Point vortices in a two-dimensional fluid may be arranged so that all resulting vortex velocities are identical; these are equivalent to force-free arrangements of 2D Coulomb charges in a homogeneous field. For generic choice of vortex circulations, the number of stationary configurations is finite, but a few infinite solution families (of arbitrary positive dimension) are known to exist. This paper extends the classes of point vortex systems known to have a finite or infinite number of stationary configurations respectively. The main tool is a new minimal system of polynomial equations characterizing these configurations. The polynomial system is extended to include symmetric vortex configurations such as vortex streets and rings, as well as circulation symmetries (vortices with identical circulations). By expressing the systems as a multilinear differential equation, one-dimensional solution sets can be constructed for a wide range of systems, including vortex streets and rings.