Permutation Groups Induced by Derksen Groups in Characteristic Two
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Permutation Groups Induced by Derksen Groups in Characteristic Two Keisuke Hakuta1 Received: 3 December 2019 / Revised: 22 May 2020 / Accepted: 1 June 2020 / © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2020
Abstract We consider the so-called Derksen group which is a subgroup of the polynomial automorphism group of the polynomial ring in n variables over a field. The Derksen group is generated by affine automorphisms and one particular non-linear automorphism. Derksen (1994) proved that if the characteristic of the underlying field is zero and n ≥ 3, then the Derksen group is equal to the entire tame subgroup. The result is called Derksen’s Theorem. It is quite natural to ask whether the same property holds for positive characteristic. In this paper, we point out that the question can be easily answered negatively when the underlying field is of characteristic two. We shall also prove that the permutation group induced by the Derksen group over a finite field of characteristic two is a proper subgroup of the alternating group on the n-dimensional linear space over the finite field. This is a stronger result that Derksen’s Theorem does not hold when the underlying field is a finite field of characteristic two. Keywords Affine algebraic geometry · Derksen group · Polynomial automorphism · Tame automorphism · Tame subgroup · Finite field · Permutation Mathematics Subject Classification (2010) 14R10 · 12E20 · 20B25
1 Introduction Let K be a field. We denote by K∗ := K \ {0} the multiplicative group of K. Throughout this paper, we use the symbol K[X1 , . . . , Xn ] (n ≥ 1) to denote the polynomial ring over K. We represent the characteristic of the field K by p := char(K). Let GAn (K) be the general automorphism group, namely, the group of automorphisms of Spec K[X1 , . . . , Xn ] over Spec K. Recall that GAn (K) is anti-isomorphic to the group of algebraic automorphisms AutK K[X1 , . . . , Xn ]. We identify GAn (K) with AutK K[X1 , . . . , Xn ] via the above correspondence. A polynomial automorphism F = (f1 , . . . , fn ) ∈ GAn (K) is said to be affine Keisuke Hakuta
[email protected] 1
Institute of Science and Engineering, Academic Assembly, Shimane University, Shimane, Japan
K. Hakuta
if degfi = 1 for all i (1 ≤ i ≤ n). The set of affine automorphisms is denoted by Affn (K). We recall that Affn (K) ∼ = Kn GLn (K). A polynomial map Ei (g) of the form Ei (g) = (X1 , . . . , Xi−1 , Xi + g, Xi+1 , . . . , Xn ),
g ∈ K[X1 , . . . , Xi−1 , Xi+1 , . . . , Xn ], (1.1) is also a polynomial automorphism (since Ei (g)−1 = Ei (−g)), and is said to be elementary. The subgroup of GAn (K) generated by the elementary automorphisms is denoted by EAn (K). The tame subgroup TAn (K) is the subgroup of GAn (K) generated by two subgroups Affn (K) and EAn (K). The Tame Generators Problem asks whether GAn (K) = TAn (K) holds, and is related to the Jacobian conjecture. A polynomial automorphism Ja,f of the form Ja,f = (a1 X1 + f1 (X2 , . . . , Xn ), a2
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