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

Maubach, Stefan, and Willems, Roel. "Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group." Serdica Mathematical Journal 37.4 (2011): 305-322. <http://eudml.org/doc/281558>.

@article{Maubach2011,
abstract = {2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.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= F2. The latter result provides a tool to show that a map is not linearizable.},
author = {Maubach, Stefan, Willems, Roel},
journal = {Serdica Mathematical Journal},
keywords = {Polynomial Automorphisms; Permutation Groups; Tame Automorphism Group; Polynomial automorphisms; permutation groups; tame automorphism group},
language = {eng},
number = {4},
pages = {305-322},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group},
url = {http://eudml.org/doc/281558},
volume = {37},
year = {2011},
}

TY - JOUR
AU - Maubach, Stefan
AU - Willems, Roel
TI - Polynomial automorphisms over finite fields: Mimicking tame maps by the Derksen group
JO - Serdica Mathematical Journal
PY - 2011
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 37
IS - 4
SP - 305
EP - 322
AB - 2010 Mathematics Subject Classification: 14L99, 14R10, 20B27.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= F2. The latter result provides a tool to show that a map is not linearizable.
LA - eng
KW - Polynomial Automorphisms; Permutation Groups; Tame Automorphism Group; Polynomial automorphisms; permutation groups; tame automorphism group
UR - http://eudml.org/doc/281558
ER -