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Number of points on a complex sphere with pairwise inner product restriction

Considered the following inner products: $(1)$ $\langle x,y \rangle = \sum_{t=1}^{n}x_{t}y_{t}$ $(2)$ $\langle x,y \rangle_{c} = \sum_{t=1}^{n}x_{t}\bar{y}_{t}$ consider the following surfaces: ...