Chebotar\"ev proved that every minor of a discrete Fourier matrix of prime order is nonzero. We prove a generalization of this result that includes analogues for discrete cosine and discrete sine matrices as special cases. We establish these results via a generalization of the Bir\'o-Meshulam-Tao uncertainty principle to functions with symmetries that arise from certain group actions, with some of the simplest examples being even and odd functions. We show that our result is best possible and in some cases is stronger than that of Bir\'o-Meshulam-Tao. Some of these results hold in certain circumstances for non-prime fields; Gauss sums play a central role in such investigations.
翻译:Chebotar\"ev" 证明, 离散的 Fourier 质谱矩阵的每个未成年人都不是零。 我们证明这一结果的概括性, 包括离散的余弦和离散的正弦矩阵的类比作为特例。 我们通过将Bir\'o- Meshulam-Tao不确定性原则概括化为某些群体行动产生的对称功能来确立这些结果, 有些最简单的例子是偶数和奇数的功能。 我们证明我们的结果是最好的, 在某些情况下比Bir\'o- Meshulam-Tao的结果更强。 其中一些结果在某些特定情况下对非主要领域是存在的; 高斯在这类调查中起着核心作用 。