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

TL;DR

This paper introduces a new framework combining fiber criteria and the AGW method to construct and verify permutation and complete permutation polynomials over finite fields, simplifying proofs and enabling explicit examples.

Contribution

It develops a general fiber-based framework for constructing permutation polynomials, extending existing results with simplified proofs and concrete examples.

Findings

New proofs of permutation polynomial results using fiber criteria

A general framework for constructing complete permutation polynomials

Explicit examples demonstrating the framework's effectiveness

Abstract

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.