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My best problem

Source: RMM 2020 Shortlist

October 15, 2021
algebra

Problem Statement

Let n>1n>1 be a positive integer and S\mathcal S be the set of nthn^{\text{th}} roots of unity. Suppose PP is an nn-variable polynomial with complex coefficients such that for all a1,…,an∈Sa_1,\ldots,a_n\in\mathcal S, P(a1,…,an)=0P(a_1,\ldots,a_n)=0 if and only if a1,…,ana_1,\ldots,a_n are all different. What is the smallest possible degree of PP?
Adam Ardeishar and Michael Ren
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