On linear relations between roots of unity

Extract

In a recent paper I. J. Schoenberg [1] considered relations

where the a_v are rational integers and the ζ_v are roots of unity. We may in (1) replace all negative coefficients a_v by −a_v replacing at the same time ζ_v by −ζ_v so that we may, if it is convenient, assume that all a_v are positive. If we do this and arrange the ζ_r so that their arguments do not decrease with v then (1) can, as suggested by Schoenberg (oral communication) be interpreted as a convex polygon with integral sides whose angles are rational when measured in degrees. Accordingly we shall call a relation (1) a polygon if all a_v are non-negative. We shall call a polygon (1) k-sided if all a_v are positive. The polygon is called degenerate if two of the ζ_v are equal. Schoenberg calls these polygons polar rational polygons (abbreviated prp) because the vectors composing them have rational coordinates in their polar representations. Schoenberg showed that every prp can be obtained as a linear combination with integral positive or negative coefficients of regular p-gons where p is a prime.