Dimension of automorphisms with fixed degree for polynomial algebras

TL;DR

This paper extends the understanding of the structure and dimension of automorphisms with fixed degree in polynomial and free associative algebras across different characteristics, revealing their constructibility and dimension properties.

Contribution

It generalizes Furter's characteristic zero result to arbitrary characteristic and to free associative algebras, establishing the constructibility and dimension of automorphisms with fixed degree.

Findings

Automorphisms with fixed degree form constructible sets.

Dimension of these sets is n+6 for degree n≥2.

Results hold across various algebraic structures and characteristics.

Abstract

Let $K[x,y]$ be the polynomial algebra in two variables over an algebraically closed field $K$. We generalize to the case of any characteristic the result of Furter that over a field of characteristic zero the set of automorphisms $(f,g)$ of $K[x,y]$ such that $\max\{\text{deg}(f),\text{deg}(g)\}=n\geq 2$ is constructible with dimension $n+6$. The same result holds for the automorphisms of the free associative algebra $K< x,y>$. We have also obtained analogues for free algebras with two generators in Nielsen -- Schreier varieties of algebras.