Classifying minimal vanishing sums of roots of unity

Abstract

A vanishing sum of roots of unity is called minimal if no proper, nonempty sub-sum of it vanishes. This paper classifies all minimal vanishing sums of roots of unity of weight at most 16 by hand, thereby uncovering new phenomena beyond the earlier 1998 classification of Poonen and Rubinstein (SIAM J. Discrete Math.) that went up to weight 12. The paper also develops an algorithm to explore higher weights up to 21, yielding a conjectural extension of the classification.