On Automorphisms of the Tame Polynomial Automorphism Group in Positive Characteristic

TL;DR

This paper proves that in positive characteristic fields, automorphisms of the tame polynomial automorphism group that fix diagonal matrices preserve all tame automorphisms up to linear inner automorphisms, for polynomial algebras in more than three variables.

Contribution

It establishes a structural preservation property of automorphisms of the tame automorphism group in positive characteristic fields, extending understanding of automorphism groups.

Findings

Automorphisms fixing diagonal matrices preserve all tame automorphisms.

The result holds for polynomial algebras in more than three variables.

The proof applies to algebraically closed fields of positive characteristic not equal to 2.

Abstract

In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every diagonal matrix preserves, up to composition with a linear inner automorphism, every tame automorphism.