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 $\Arg(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$.
Deep Dive into A symmetry property of some harmonic algebraic curves.
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 $\Arg(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$.
Let z = {z 1 , . . . , z n } be a multiset of n points in the complex plane C and P the monic polynomial with root set z:
For θ a fixed real number of your choice, consider
The set C θ (P ) coincides up to z, to the set {z ∈ C : Arg(P (z)) = θ[π]}. These curves arise in the Gauss approach to the Fundamental Theorem of Algebra (see e.g. Stillwell [3], and Martin & al. [1]). In their paper Martin & al. [1] and then Savitt [2] initiated the study of the combinatorial topology of the families C θ (P ). The idea are the following ones: the curves C θ (P ) have 2n asymptotes at angles (πk + θ)/n, for k ∈ {0, . . . , 2n -1}, and form in the generic case n non intersecting curves. This induces a matching: k and k ′ are matched if and only if the asymptotes (πk + θ)/n and (πk ′ + θ)/n lie on the same connected component in C θ (P ). The papers [1] and [2] aim at studying these matchings, and also the properties of the so-called necklaces, formed by the families of matchings obtained when θ traverses the set [0, π].
Let us now state and prove our result. The set z is clearly included in C θ (P ). It turns out that when z is included in the unit circle U = {z : |z| = 1}, the set C θ (P ) ∩ U presents a quite surprising symmetry -illustrated at Figure 1 -that can be stated as follows.
where G(z) is the regular n-gon on U, with set of vertices e i(Ω+2kπ/n) , k = 1, . . . , n , for
There exists a purely geometric proof of this Proposition using that the measure of a central angle is twice that of the inscribed angle intercepting the same arc; we provide below a more compact analytic proof.
Proof. We will only consider z / ∈ z. We have the equivalence:
where Arg(z) ∈ R/2πZ stands for (any chosen determination of) the argument of z = 0. Now for any ν and ψ real numbers,
Hence, z ∈ C θ (P ) \ z is equivalent to:
which leads to the conclusion at once. Note. If z i is a root of multiplicity k of P , and if z i belongs to G(z), then in the neighborhood of z i , C θ (P ) has k tangents, one of them coinciding with the tangent of the circle at z i . Moreover, it is simple to check that if z i is not on G(z), then the tangents of C θ (P ) at z i are not tangent to U.
This work has been supported by the ANR project MARS (BLAN06-2 0193).
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