Finiteness of Fixed Equilibrium Configurations of Point Vortices in the Plane with Background Flow

TL;DR

This paper establishes a finite upper bound on the number of fixed equilibrium configurations of point vortices in a plane with background flow, using algebraic geometry tools to transform and analyze the system.

Contribution

It introduces a new physically meaningful definition of equilibrium configurations and derives a generic upper bound on their number using polynomial system transformation and algebraic geometry.

Findings

Finite upper bound on equilibrium configurations derived

Transformation of vortex system into polynomial form for analysis

Application of BKK theory and Bézout's theorem to establish finiteness

Abstract

For a dynamic system consisting of $n$ point vortices in an ideal plane fluid with a steady, incompressible and} irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial $w$ that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound \smash{$\fr{(m+n-1)!}{(m-1)!\,n_1!\cdots n_{i_0}!}$} for the number of fixed equilibrium configurations. Here, $m=\deg w$, $i_0$ is the number of species, and each $n_i$ is the number of vortices in a species. We transform the rational function system arisen from} fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after D. N. Bernshtein, A. G. Khovanskii and A. G. Kushnirenko) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from B\'ezout's theorem or the BKK root count by T. Y. Li and X.-S. Wang.}