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Article overview
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Finiteness of Fixed Equilibrium Configurations of Point Vortices in the Plane with Background Flow | Pak-Leong Cheung
; Tuen Wai Ng
; | Date: |
8 Sep 2014 | 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.} | Source: | arXiv, 1409.2284 | Services: | Forum | Review | PDF | Favorites |
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