The Subpower Membership Problem for Mal'cEV Algebras

Given tuples a1, . . . , ak and b in A n for some algebraic structure A, the subpower membership problem asks whether b is in the subalgebra of An that is generated by a1, . . . , ak . For A a finite group, there is a folklore algorithm which decides this problem in time polynomial in n and k. We show that the subpower membership problem for any finite Mal’cev algebra is in NP and give a polynomial time algorithm for any finite Mal’cev algebra with finite signature and prime power size that has a nilpotent reduct. In particular, this yields a polynomial algorithm for finite rings, vector spaces, algebras over fields, Lie rings and for nilpotent loops of prime power order.