Automorphisms of polynomial algebras and Dirichlet series
Vesselin Drensky, Jie-Tai Yu

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.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions · Mathematical Dynamics and Fractals
