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

Abstract

If F is a polynomial automorphism over a finite field Fq in dimension n, then it induces a permutation pqr(F) of (Fqr)n for every r О N*. We say that F can be "mimicked" by elements of a certain group of automorphisms G if there are gr О G such that pqr(gr) = pqr(F). Derksen's theorem in characteristic zero states that the tame automorphisms in dimension n і 3 are generated by the affine maps and the one map (x1+x22, x2,ј, xn). We show that Derksen's theorem is not true in characteristic p in general. However, we prove a modified, weaker version of Derksen's theorem over finite fields: we introduce the Derksen group DAn(Fq), n і 3, which is generated by the affine maps and one well-chosen nonlinear map, and show that DAn(Fq) mimicks any element of TAn(Fq). Also, we do give an infinite set E of non-affine maps which, together with the affine maps, generate the tame automorphisms in dimension 3 and up. We conjecture that such a set E cannot be finite. We consider the subgroups GLINn(k) and GTAMn(k). We prove that for k a finite field, these groups are equal if and only if k\not = F2. The latter result provides a tool to show that a map is not linearizable.