2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163655Let 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.13 pages, revised version, new results are addedCommutative AlgebraCombinatoricsRings and Algebras11T06; 13B25; 05A15; 30B50; 16S10; 17A50.text