Bilinear relative equilibria of identical point vortices

A new class of bilinear relative equilibria of identical point vortices in which the vortices are constrained to be on two perpendicular lines, taken to be the $x$- and $y$-axes of a cartesian coordinate system, is introduced and studied. In general we have $m$ vortices on the $y$-axis and $n$ on the $x$- axis. We define generating polynomials $q(z)$ and $p(z)$, respectively, for each set of vortices. A second order, linear ODE for $p(z)$ given $q(z)$ is derived. Several results relating the general solution of the ODE to relative equilibrium configurations are established. Our strongest result, obtained using Sturm's comparison theorem, is that if $p(z)$ satisfies the ODE for a given $q(z)$ with its imaginary zeros symmetric relative to the $x$-axis, then it must have at least $n-m+2$ simple, real zeros. For $m=2$ this provides a complete characterization of all zeros, and we study this case in some detail. In particular, we show that given $q(z) = z^2 + \eta^2$, where $\eta$ is real, there is a unique $p(z)$ of degree $n$, and a unique value of $\eta^2 = A_n$, such that the zeros of $q(z)$ and $p(z)$ form a relative equilibrium of $n+2$ point vortices. We show that $A_n \approx \frac{2}{3}n + \frac{1}{2}$, as $n\rightarrow\infty$, where the coefficient of $n$ is determined analytically, the next order term numerically.