A symmetry property of some harmonic algebraic curves
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The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that $\Arg(P(z))=θ$, form a regular $n$-gon, where $n$ is the degree of $P$.