与分圆方程有关的差集的不可能性(英文)

Impossibility of Difference Sets Depending on Certain Cyclotomic Equation

设ｆ＝Ｏｒｄｐ（ｑ），ξｐ为本原ｐ次单位根，γ∈Ｚ［ξｐ］。若ｆ是奇数且γγ＝ｎ（其中ｎ∈Ｚ，ｑ｜ｎ），加上别的一些条件，Ａｒａｓｕ和Ｐｏｔ证明了γ具有形式：γ＝∑ｅｉ＝１ａｉ∑ｆ－１ｊ＝０ξｉｑｊｐ（其中ｐ－１＝ｅｆ），并且指出如果条件“ｆ是奇数”能去掉的话，将是一个很好的结果．本文作者研究了ｆ为偶数的情形并得到相应的结论。

Let f= Or d p(q), ξ p be a primitive p th root of unity, γ∈Z ξ p . Suppose that f is odd and γ=n (where n∈Z,q|n ). By some assumption, K.T.Arasu and A.Pott obtain that γ can be written as the form γ=∑ e i=1 a i∑ f-1 j=0 ξ iq j p(where p-1=ef ). And note that it would be nice if the condition “ f is odd” could be removed. In this paper, the author will deal with the case “ f is even” and get a corresponding result.