A unifying theoretical framework is presented, in which the connections among Costas sequences, circular Costas sequences, Costas polynomials, the shifting property, and Welch sequences are extended to the multidimensional context. Several conjectures on multidimensional periodic Costas arrays by J. Ortiz-Ubarri et al. are proved. Furthermore, a conjecture on Costas polynomials over extension fields presented by Muratovic-Ribic et al. is showed to be a multidimensional extension of a conjecture by Golomb and Moreno on circular Costas sequences. A weaker version of said conjecture is proved by considering a multidimensional extension of the shifting Costas property defined by O. Moreno.
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