Mathematics Colloquia and Seminars

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Take a natural number n and let t be an nth primitive root of unity. A vector (a₀, a₁, ..., a_n-1) is a "vanishing sum of n-th roots of unity" if a₀ + a₁t + a₂t ² + ... + a_n-1t ^n-1 = 0. The set of all such vectors is a subspace of Rⁿ. In this talk we will give a basis for this space (first established by DeBruijn, Redei and Schoenberg) as well as a basis for the orthogonal complement. An element of the space is "minimal" if it has nonnegative integer entries and is not the sum of two other nonzero, nonnegative integer elements. We will show how to construct minimal sums with arbitrarily large entries.