Euler Configurations and Quasi-Polynomial Systems

Consider the problem of three point vortices (also called Helmholtz' vortices) on a plane, with arbitrarily given vorticities. The interaction between vortices is proportional to $1/ r$, where $r$ is the distance between two vortices. The problem has 2 equilateral and at most 3 collinear normalized relative equilibria. This 3 is the optimal upper bound. Our main result is that the above standard statements remain unchanged if we consider an interaction proportional to $r^b$, for any $b<0$. For $0 < b < 1$, the optimal upper bound becomes 5. For positive vorticities and any $b<1$, there are exactly 3 collinear normalized relative equilibria. The case $b=-2$ of this last statement is the well-known theorem due to Euler: in the Newtonian 3-body problem, for any choice of the 3 masses, there are 3 Euler configurations (also known as the 3 Euler points). These small upper bounds strengthen the belief of Kushnirenko and Khovanskii [18]: real varieties defined by simple systems should have a simple topology. We indicate some hard conjectures about the configurations of relative equilibrium and suggest they could be attacked within the quasi-polynomial framework.