Note on vanishing power sums of roots of unity

2015/03/25 by Neeraj Kumar, Krishan Kumar, Kumar, Neeraj +1
Mathematics · #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #FOS: Mathematics #Graph theory and applications #Number Theory (math.NT) #Primary 11L03 #Secondary 11R18

paper · pdf · doi:10.48550/arxiv.1503.07281

openalex publication_date 2015/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a given positive integers m and ℓ, we give a complete list of positive integers n for which their exist mth roots of unity x₁,…,x_n ∈ ℂ such that x₁^ℓ + ⋯ + x_n^ℓ=0. This extends the earlier result of Lam and Leung on vanishing sums of roots of unity. Furthermore, we prove that for which integers n with 2 ≤ n ≤ m, there are distinct mth roots of unity x₁,…,x_n ∈ ℂ such that x₁^ℓ + ⋯ + x_n^ℓ=0.

Citations

• Prime ideals and regular sequences of symmetric polynomials

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