On Permutation Trinomials and Complete Permutation Polynomials via Fiber Criteria over Finite Fields

We give new, short proofs of recent permutation polynomial results of Bousalmi, Bayad, and Derbal by reducing the verification to explicit computations on a three-element multiplicative subgroup via Zieve's fiber criterion. Building on this approach, we develop a general framework -- combining Zieve's theorem with the AGW criterion -- for constructing complete permutation polynomials over finite fields through a fiber decomposition over the cube roots of unity. A scalar specialization of the criterion yields families that are easy to produce and verify. We illustrate the construction with concrete examples and show through counterexamples that the underlying arithmetic conditions are sharp.