Polynomial automorphisms over finite fields: Mimicking non-tame and tame maps by the Derksen group

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If F is a polynomial automorphism over a finite field in dimension n, then it induces a bijection πqᵣ(F) of (ᵣ)ⁿ for every r∈. We say that F can be `mimicked' by elements of a certain group of automorphisms G if there are gᵣ∈ G such that πqᵣ(gᵣ)=πqᵣ(F). We show that the Nagata automorphism (and any other automorphism in three variables fixing one variable) can be mimicked by tame automorphisms. This on the one hand removes the hope of showing that such an automorphism is non-tame by studying one bijection it induces, but on the other hand indicates that the bijections of tame automorphisms coincide with bijections of all automorphisms. In section 5 we show that the whole tame group ₙ() in turn can be mimicked by the Derksen subgroup ₙ(), which is the subgroup generated by affine maps and one particular element. It is known that if a field k has characteristic zero, then ₙ(k)=ₙ(k), which is not expected to be true for characteristic p. In section 6 we consider the subgroups ₙ(k) and ₙ(k) of the polynomial automorphism group (which are candidates to equal the entire automorphism group). We show that ₙ()=ₙ() if q = 2, and ₙ(₂) ₙ(₂).