Dimension of automorphisms with fixed degree for free algebras of Nielsen-Schreier varieties

Abstract

Let ℳ be a Nielsen-Schreier variety of algebras over an algebraically closed field and let F(x, y) be an ℳ-free algebra on two generators. Assume that the sequence of dimensions c n of the homogeneous polynomials of degree n in F(x) satisfies the inequalities 1 = c 1 = c 2 ≤ c 3 ≤ ···. We establish that the set of automorphisms (f, g) of F(x, y) such that n = max{deg(f), deg(g)} forms a constructible set with dimension d n = c 1 + c 2+ ··· + c n + 4 + 2E, where E = 1 for unital algebras and E = 0 for nonunital algebras. The dimension of the set of coordinates f of degree n is equal to c 1 + c 2 + ··· + c n + 2 + E.