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Minimal vanishing sums of roots of unity with large coefficients
Author(s) -
Steinberger John P.
Publication year - 2008
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdn006
Subject(s) - mathematics , root of unity , zero (linguistics) , combinatorics , sums of powers , root (linguistics) , physics , philosophy , linguistics , quantum mechanics , quantum
A vanishing sum a 0 + a 1 ζ n + … + a a − 1ζ n n − 1 = 0 , where ζ n is a primitive n th root of unity and the a i s are non‐negative integers is called minimal if the coefficient vector ( a 0 , … , a n −1 ) does not properly dominate the coefficient vector of any other such non‐zero sum. We show that for every c ∈ℕ there is a minimal vanishing sum of n th roots of unity with its greatest coefficient equal to c , where n is of the form 3 pq for odd primes p , q . This solves an open problem posed by Lenstra Jr.