Hardness results for the subpower membership problem

The main result of this paper shows that if [Formula: see text] is a consistent strong linear Maltsev condition which does not imply the existence of a cube term, then for any finite algebra [Formula: see text] there exists a new finite algebra [Formula: see text] which satisfies the Maltsev condition [Formula: see text], and whose subpower membership problem is at least as hard as the subpower membership problem for [Formula: see text]. We characterize consistent strong linear Maltsev conditions which do not imply the existence of a cube term, and show that there are finite algebras in varieties that are congruence distributive and congruence [Formula: see text]-permutable ([Formula: see text]) whose subpower membership problem is EXPTIME-complete.

SOLVABILITY OF SYSTEMS OF POLYNOMIAL EQUATIONS OVER FINITE ALGEBRAS

We study the algorithmic complexity of determining whether a system of polynomial equations over a finite algebra admits a solution. We prove that the problem has a dichotomy in the class of finite groupoids with an identity element. By developing the underlying idea further, we present a dichotomy theorem in the class of finite algebras that admit a non-trivial idempotent Maltsev condition. This is a substantial extension of most of the earlier results on the topic.