The profinite polynomial automorphism group

Abstract

We introduce an extension of the (tame) polynomial automorphism group over finite fields: the profinite (tame) polynomial automorphism group, which is obtained by putting a natural topology on the automorphism group. We show that most known candidate non-tame automorphisms are inside the profinite tame polynomial automorphism group, giving another result showing that tame maps are potentially "dense" inside the set of automorphisms. We study the profinite tame automorphism group and show that it is not far from the set of bijections obtained by endomorphisms.

A new formulation of the Jacobian Conjecture in characteristic $p$

“…The reason for this is that (1) any affine or triangular map having determinant Jacobian 1 has a preimage under π, (2) any tame automorphism of determinant Jacobian 1 can indeed be written as a composition of affine and triangular automorphisms of determinant Jacobian 1. (See [1] lemma 3.4.) Now an obvious question is whether the map π : SA n (Z) −→ SA n (F p ) is surjective or not; this question is interesting as nonsurjectivity would yield non-tame maps due to the above remark.…”

“…For each of these sets, we define ME d n (R) = {F ∈ ME n (R) | deg(F ) ≤ d}, GA d (R) = GA n (R) ∩ ME d n (R) etc. We use the notation x α = x α 1 1 · · · x αn n if α ∈ N n .…”

“…The reason for this is that (1) any affine or triangular map having determinant Jacobian 1 has a preimage under π, (2) any tame automorphism of determinant Jacobian 1 can indeed be written as a composition of affine and triangular automorphisms of determinant Jacobian 1. (See [1]…”

“…Now an obvious question is whether the map π : SA n (Z) −→ SA n (F p ) is surjective or not; this question is interesting as nonsurjectivity would yield non-tame maps due to the above remark. This is part of the topic of the papers [1,2,3]. Definition 5.2.…”

On Permutations Induced by Tame Automorphisms Over Finite Fields

“…Thus, Question 1 is a quite important problem for the Tame Generators Problem in positive characteristic. Furthermore, we refer the reader to [5,Section 1.2] for several questions related to [3,Theorem 2.3]. The present paper deals with permutations induced by tame automorphisms over finite fields.…”