AI 中文总结
本文研究循环n次根的数量与傅里叶支撑集不交性，证明当n为合数时存在时域和频域支撑集不交的向量对，修正了Haagerup针对素数n的证明简化步骤的适用范围。
AI 中文摘要
循环n次根是满足特定多元齐次多项式方程组的n维复向量。单模循环n次根与首项为1的双单模向量（CAZAC序列）之间存在一一对应关系。Björck和Saffari猜想，循环n次根的集合有限当且仅当n是无平方因子数。已知当n不是无平方因子数时，该集合是无限的；当n是素数时，该集合是有限的。Haagerup关于素数n的证明中的关键简化步骤是，证明循环n次根的无限性（对任意n）蕴含存在两个在时域和频域均具有不交支撑集的向量。本文证明，当n是合数时，这样的向量对总是存在，这表明原始的简化步骤对于无平方因子的合数情形并不适用。文中还讨论了支撑集与其傅里叶变换不交的单个向量的存在性问题。
英文摘要
A cyclic $n$-root is an $n$-dimensional complex vector that solves a particular set of multivariate homogeneous polynomial equations. There is a one-to-one correspondence between unimodular cyclic $n$-roots and bi-unimodular vectors (CAZAC sequences) with leading entry one. It was conjectured by Björck and Saffari that the set of cyclic $n$-roots is finite if and only if $n$ is square free. It is known that such a set is infinite if $n$ is not square free, and finite if $n$ is prime. A critical reduction in Haagerup's proof for prime $n$ is to show that infinity of cyclic $n$-roots (for any $n$) implies existence of two vectors with disjoint supports in both the time domain and the frequency domain. In this paper we show that such pair of vectors always exist if $n$ is composite, indicating that the original reduction is not adequate for composite square free cases. A discussion on the existence of a single vector whose support is disjoint with its Fourier transform is also included.
CommentsThanks to AI, I fixed a few typos in formulas and corrected several imprecise claims
