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Places where |p(z)|=1

Source: Miklós Schweitzer 2017, problem 5

January 13, 2018
algebrapolynomial

Problem Statement

For every non-constant polynomial pp, let Hp={z∈C ∣ ∣p(z)∣=1}H_p=\big\{z\in \mathbb{C} \, \big| \, |p(z)|=1\big\}. Prove that if Hp=HqH_p=H_q for some polynomials p,qp,q, then there exists a polynomial rr such that p=rmp=r^m and q=ξ⋅rnq=\xi\cdot r^n for some positive integers m,nm,n and constant ∣ξ∣=1|\xi|=1.
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