Polynomial endomorphisms over finite fields: experimental results

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Given a finite field and n∈, one could try to compute all polynomial endomorphisms ⁿ ⁿ up to a certain degree with a specific property. We consider the case n=3. If the degree is low (like 2,3, or 4) and the finite field is small (q≤ 7) then some of the computations are still feasible. In this article we study the following properties of endomorphisms: being a bijection of ⁿ ⁿ, being a polynomial automorphism, being a Mock automorphism, and being a locally finite polynomial automorphism. In the resulting tables, we point out a few interesting objects, and pose some interesting conjectures which surfaced through our computations.