Automorphisms of polynomial algebras and Dirichlet series

Details Automorphisms of polynomial algebras and Dirichlet series Automorphisms of polynomial algebras and Dirichlet series

2008-06-04

arxiv.org/abs/0806.0681v2

Mathematics

Commutative Algebra

13 pages, revised version, new results are added

Scientific paper

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). We have also obtained analogues for free algebras with two generators in Nielsen - Schreier varieties of algebras.

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