Automorphisms of polynomial algebras and Dirichlet series
Commutative Algebra
2008-06-27 v2 Combinatorics
Rings and Algebras
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.
Keywords
Cite
@article{arxiv.0806.0681,
title = {Automorphisms of polynomial algebras and Dirichlet series},
author = {Vesselin Drensky and Jie-Tai Yu},
journal= {arXiv preprint arXiv:0806.0681},
year = {2008}
}
Comments
13 pages, revised version, new results are added