OFFSET
0,8
COMMENTS
In this context, a relation of weight n is a multiset of n roots of unity which sum to zero, and it is minimal if no proper nonempty sub-multiset sums to zero. Relations are rotationally equivalent if they are obtained by multiplying each element by a common root of unity.
Mann classified the minimal relations up to weight 7, Conway and Jones up to weight 9, and Poonen and Rubinstein up to weight 12.
LINKS
J. H. Conway and A. J. Jones, Trigonometric diophantine equations (On vanishing sums of roots of unity), Acta Arithmetica 30(3), 229-240 (1976).
Henry B. Mann, On linear relations between roots of unity, Mathematika 12(2), 107-117 (1965).
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Math. 11(1), 135-156 (1998). Also at arXiv:math/9508209 [math.MG] with some typos corrected.
EXAMPLE
Writing e(x) = exp(2*Pi*i*x), then e(1/6)+e(1/5)+e(2/5)+e(3/5)+e(4/5)+e(5/6) = 0 and this is the unique (up to rotation) minimal relation of weight 6.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Christopher E. Thompson, May 15 2016
STATUS
approved