On linear relations between roots of unity
On linear relations between roots of unity
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单位根之间的线性关系
DOI：
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发表时间：
1965
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通讯作者：
H. B. Mann
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文献类型：
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作者：
H. B. Mann
在最近的一篇论文中，I. J. Schoenberg[1]考虑了a v是有理数，ζ v是单位根的关系。我们可以在(1)中用- av替换所有的负系数同时用- ζ v替换ζ v这样我们就可以，如果方便的话，假设所有的a v都是正的。如果我们这样做，并安排ζ r，使它们的参数不随v减小，那么(1)可以，正如勋伯格（口头交流）所建议的那样，被解释为一个凸多边形，它的边是积分边，当用度来测量时，角是有理的。因此，如果所有的v都是非负的，我们称关系(1)为多边形。如果所有的a v都是正的，我们称一个多边形为k边。如果两个ζ v相等这个多边形就被称为简并多边形。勋伯格称这些多边形为极有理多边形（简称prp），因为构成它们的向量在它们的极坐标表示中具有有理坐标。Schoenberg证明了每一个prp都可以用正则p -gons的正或负积分系数的线性组合得到，其中p是素数。
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.