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- W4298419952 abstract "We prove some basic theorems concerning lemniscate configurations in an Euclidean space of dimension $ n geq 3$. Lemniscates are defined as follows. Given m points $w_j $ in $mathbb R^n$, consider the function $F(x)$ which is the product of the distances $ |x-w_j|$: the singular level sets of the function $F$ are called lemniscates. We show via complex analysis that the critical points of $F$ have Hessian of positivity at least $(n-1)$. This implies that, if $F$ is a Morse function, then $F$ has only local minima and saddle points with negativity 1. The critical points lie in the convex span of the points $|w_j| $ (these are absolute minima): but we made also the discovery that $F$ can also have other local minima, and indeed arbitrarily many. We discuss several explicit examples. We finally prove in the appendix that all critical points are isolated." @default.
- W4298419952 created "2022-10-02" @default.
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- W4298419952 date "2015-06-05" @default.
- W4298419952 modified "2023-09-30" @default.
- W4298419952 title "Higher dimensional Lemniscates: the geometry of $r$ particles in $n$-space with logarithmic potentials" @default.
- W4298419952 doi "https://doi.org/10.48550/arxiv.1506.01919" @default.
- W4298419952 hasPublicationYear "2015" @default.
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