Bounding the free spectrum of nilpotent algebras of prime power order

Abstract

Let $\mathbf{A}$ be a finite nilpotent algebra in a congruence modular variety with finitely many fundamental operations. If $\mathbf{A}$ is of prime power order, then it is known that there is a polynomial $p$ such that for every $n \in \mathbb{N}$, every $n$-generated algebra in the variety generated by $\mathbf{A}$ has at most $2^{p(n)}$ elements. We present a bound on the degree of this polynomial.