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.