AI summaryⓘ
The authors study special complex vectors called cyclic n-roots that satisfy certain polynomial equations. They focus on a conjecture linking the finiteness of these roots to whether n is square-free (meaning n has no repeated prime factors). While it’s known that the set of these roots is infinite if n is not square-free and finite for prime n, the authors find that a key technique used in a previous proof for prime n fails for composite square-free n. Specifically, they show that certain pairs of vectors with non-overlapping supports in time and frequency always exist when n is composite, suggesting this approach cannot settle the conjecture in those cases. They also discuss related vectors disjoint from their Fourier transforms.
cyclic n-rootcomplex vectorunimodularbi-unimodular vectorsquare-free integerFourier transformsupport of a vectorCAZAC sequencespolynomial equationsvector time domain
Abstract
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.