Automorphisms of polynomial algebras and Dirichlet series

TL;DR

This paper provides exact formulas and asymptotic analysis for counting automorphisms of polynomial and free associative algebras over finite fields, along with their Dirichlet series generating functions.

Contribution

It introduces new formulas and asymptotics for automorphism counts and extends results to free algebras in Nielsen-Schreier varieties.

Findings

Derived exact formulas for automorphism counts.

Established asymptotic behavior of automorphism numbers.

Presented Dirichlet series generating functions for automorphisms.

Abstract

Let GF(q)[x,y] be the polynomial algebra in two variables over the finite field GF(q) with q elements. We give an exact formula and the asymptotics for the number p(n) of automorphisms (f,g) of GF(q)[x,y] such that max{deg(f),deg(g)}=n. We describe also the Dirichlet series generating function p(1)/1^s+p(2)/2^s+p(3)/3^s+.... The same results hold for the automorphisms of the free associative algebra GF(q)<x,y>. We have also obtained analogues for free algebras with two generators in Nielsen - Schreier varieties of algebras.